### 56 Rings

This chapter deals with domains that are additive groups (see IsAdditiveGroup (55.1-6) closed under multiplication *. Such a domain, if * and + are distributive, is called a ring in GAP. Each division ring, field (see 58), or algebra (see 62) is a ring. Important examples of rings are the integers (see 14) and matrix rings.

In the case of a ring-with-one, additional multiplicative structure is present, see IsRingWithOne (56.3-1). There is a little support in GAP for rings that have no additional structure: it is possible to perform some computations for small finite rings; infinite rings are handled by GAP in an acceptable way in the case that they are algebras.

Also, the SONATA package provides support for near-rings, and a related functionality for multiplicative semigroups of near-rings is available in the Smallsemi package.

Several functions for ring elements, such as IsPrime (56.5-8) and Factors (56.5-9), are defined only relative to a ring R, which can be entered as an optional argument; if R is omitted then a default ring is formed from the ring elements given as arguments, see DefaultRing (56.1-3).

#### 56.1 Generating Rings

##### 56.1-1 IsRing
 ‣ IsRing( R ) ( filter )

A ring in GAP is an additive group (see IsAdditiveGroup (55.1-6)) that is also a magma (see IsMagma (35.1-1)), such that addition + and multiplication * are distributive, see IsDistributive (56.4-5).

The multiplication need not be associative (see IsAssociative (35.4-7)). For example, a Lie algebra (see 64) is regarded as a ring in GAP.

##### 56.1-2 Ring
 ‣ Ring( r, s, ... ) ( function )
 ‣ Ring( coll ) ( function )

In the first form Ring returns the smallest ring that contains all the elements r, s, ... In the second form Ring returns the smallest ring that contains all the elements in the collection coll. If any element is not an element of a ring or if the elements lie in no common ring an error is raised.

Ring differs from DefaultRing (56.1-3) in that it returns the smallest ring in which the elements lie, while DefaultRing (56.1-3) may return a larger ring if that makes sense.

gap> Ring( 2, E(4) );
<ring with 2 generators>


##### 56.1-3 DefaultRing
 ‣ DefaultRing( r, s, ... ) ( function )
 ‣ DefaultRing( coll ) ( function )

In the first form DefaultRing returns a ring that contains all the elements r, s, ... etc. In the second form DefaultRing returns a ring that contains all the elements in the collection coll. If any element is not an element of a ring or if the elements lie in no common ring an error is raised.

The ring returned by DefaultRing need not be the smallest ring in which the elements lie. For example for elements from cyclotomic fields, DefaultRing may return the ring of integers of the smallest cyclotomic field in which the elements lie, which need not be the smallest ring overall, because the elements may in fact lie in a smaller number field which is itself not a cyclotomic field.

(For the exact definition of the default ring of a certain type of elements, look at the corresponding method installation.)

DefaultRing is used by ring functions such as Quotient (56.1-9), IsPrime (56.5-8), Factors (56.5-9), or Gcd (56.7-1) if no explicit ring is given.

Ring (56.1-2) differs from DefaultRing in that it returns the smallest ring in which the elements lie, while DefaultRing may return a larger ring if that makes sense.

gap> DefaultRing( 2, E(4) );
GaussianIntegers


##### 56.1-4 RingByGenerators
 ‣ RingByGenerators( C ) ( operation )

RingByGenerators returns the ring generated by the elements in the collection C, i. e., the closure of C under addition, multiplication, and taking additive inverses.

gap> RingByGenerators([ 2, E(4) ]);
<ring with 2 generators>


##### 56.1-5 DefaultRingByGenerators
 ‣ DefaultRingByGenerators( coll ) ( operation )

For a collection coll, returns a default ring in which coll is contained.

gap> DefaultRingByGenerators([ 2, E(4) ]);
GaussianIntegers


##### 56.1-6 GeneratorsOfRing
 ‣ GeneratorsOfRing( R ) ( attribute )

GeneratorsOfRing returns a list of elements such that the ring R is the closure of these elements under addition, multiplication, and taking additive inverses.

gap> R:=Ring( 2, 1/2 );
<ring with 2 generators>
gap> GeneratorsOfRing( R );
[ 2, 1/2 ]


##### 56.1-7 Subring
 ‣ Subring( R, gens ) ( function )
 ‣ SubringNC( R, gens ) ( function )

returns the ring with parent R generated by the elements in gens. When the second form, SubringNC is used, it is not checked whether all elements in gens lie in R.

gap> R:= Integers;
Integers
gap> S:= Subring( R, [ 4, 6 ] );
<ring with 1 generators>
gap> Parent( S );
Integers


##### 56.1-8 ClosureRing
 ‣ ClosureRing( R, r ) ( operation )
 ‣ ClosureRing( R, S ) ( operation )

For a ring R and either an element r of its elements family or a ring S, ClosureRing returns the ring generated by both arguments.

gap> ClosureRing( Integers, E(4) );
<ring-with-one, with 2 generators>


##### 56.1-9 Quotient
 ‣ Quotient( [R, ]r, s ) ( operation )

Quotient returns the quotient of the two ring elements r and s in the ring R, if given, and otherwise in their default ring (see DefaultRing (56.1-3)). It returns fail if the quotient does not exist in the respective ring.

(To perform the division in the quotient field of a ring, use the quotient operator /.)

gap> Quotient( 2, 3 );
fail
gap> Quotient( 6, 3 );
2


#### 56.2 Ideals of Rings

A left ideal in a ring R is a subring of R that is closed under multiplication with elements of R from the left.

A right ideal in a ring R is a subring of R that is closed under multiplication with elements of R from the right.

A two-sided ideal or simply ideal in a ring R is both a left ideal and a right ideal in R.

So being a (left/right/two-sided) ideal is not a property of a domain but refers to the acting ring(s). Hence we must ask, e. g., IsIdeal( R, I ) if we want to know whether the ring I is an ideal in the ring R. The property IsTwoSidedIdealInParent (56.2-3) can be used to store whether a ring is an ideal in its parent.

(Whenever the term "Ideal" occurs in an identifier without a specifying prefix "Left" or "Right", this means the same as "TwoSidedIdeal". Conversely, any occurrence of "TwoSidedIdeal" can be substituted by "Ideal".)

For any of the above kinds of ideals, there is a notion of generators, namely GeneratorsOfLeftIdeal (56.2-8), GeneratorsOfRightIdeal (56.2-9), and GeneratorsOfTwoSidedIdeal (56.2-7). The acting rings can be accessed as LeftActingRingOfIdeal (56.2-10) and RightActingRingOfIdeal (56.2-10), respectively. Note that ideals are detected from known values of these attributes, especially it is assumed that whenever a domain has both a left and a right acting ring then these two are equal.

Note that we cannot use LeftActingDomain (57.1-11) and RightActingDomain here, since ideals in algebras are themselves vector spaces, and such a space can of course also be a module for an action from the right. In order to make the usual vector space functionality automatically available for ideals, we have to distinguish the left and right module structure from the additional closure properties of the ideal.

Further note that the attributes denoting ideal generators and acting ring are used to create ideals if this is explicitly wanted, but the ideal relation in the sense of IsTwoSidedIdeal (56.2-3) is of course independent of the presence of the attribute values.

Ideals are constructed with LeftIdeal (56.2-1), RightIdeal (56.2-1), TwoSidedIdeal (56.2-1). Principal ideals of the form x * R, R * x, R * x * R can also be constructed with a simple multiplication.

Currently many methods for dealing with ideals need linear algebra to work, so they are mainly applicable to ideals in algebras.

##### 56.2-1 TwoSidedIdeal
 ‣ TwoSidedIdeal( R, gens[, "basis"] ) ( function )
 ‣ Ideal( R, gens[, "basis"] ) ( function )
 ‣ LeftIdeal( R, gens[, "basis"] ) ( function )
 ‣ RightIdeal( R, gens[, "basis"] ) ( function )

Let R be a ring, and gens a list of collection of elements in R. TwoSidedIdeal, LeftIdeal, and RightIdeal return the two-sided, left, or right ideal, respectively, I in R that is generated by gens. The ring R can be accessed as LeftActingRingOfIdeal (56.2-10) or RightActingRingOfIdeal (56.2-10) (or both) of I.

If R is a left F-module then also I is a left F-module, in particular the LeftActingDomain (57.1-11) values of R and I are equal.

If the optional argument "basis" is given then gens are assumed to be a list of basis vectors of I viewed as a free F-module. (This is mainly applicable to ideals in algebras.) In this case, it is not checked whether gens really is linearly independent and whether gens is a subset of R.

Ideal is simply a synonym of TwoSidedIdeal.

gap> R:= Integers;;
gap> I:= Ideal( R, [ 2 ] );
<two-sided ideal in Integers, (1 generators)>


##### 56.2-2 TwoSidedIdealNC
 ‣ TwoSidedIdealNC( R, gens[, "basis"] ) ( function )
 ‣ IdealNC( R, gens[, "basis"] ) ( function )
 ‣ LeftIdealNC( R, gens[, "basis"] ) ( function )
 ‣ RightIdealNC( R, gens[, "basis"] ) ( function )

The effects of TwoSidedIdealNC, LeftIdealNC, and RightIdealNC are the same as TwoSidedIdeal (56.2-1), LeftIdeal (56.2-1), and RightIdeal (56.2-1), respectively, but they do not check whether all entries of gens lie in R.

##### 56.2-3 IsTwoSidedIdeal
 ‣ IsTwoSidedIdeal( R, I ) ( operation )
 ‣ IsLeftIdeal( R, I ) ( operation )
 ‣ IsRightIdeal( R, I ) ( operation )
 ‣ IsTwoSidedIdealInParent( I ) ( property )
 ‣ IsLeftIdealInParent( I ) ( property )
 ‣ IsRightIdealInParent( I ) ( property )

The properties IsTwoSidedIdealInParent etc., are attributes of the ideal, and once known they are stored in the ideal.

gap> A:= FullMatrixAlgebra( Rationals, 3 );
( Rationals^[ 3, 3 ] )
gap> I:= Ideal( A, [ Random( A ) ] );
<two-sided ideal in ( Rationals^[ 3, 3 ] ), (1 generators)>
gap> IsTwoSidedIdeal( A, I );
true


##### 56.2-4 TwoSidedIdealByGenerators
 ‣ TwoSidedIdealByGenerators( R, gens ) ( operation )
 ‣ IdealByGenerators( R, gens ) ( operation )

TwoSidedIdealByGenerators returns the ring that is generated by the elements of the collection gens under addition, multiplication, and multiplication with elements of the ring R from the left and from the right.

R can be accessed by LeftActingRingOfIdeal (56.2-10) or RightActingRingOfIdeal (56.2-10), gens can be accessed by GeneratorsOfTwoSidedIdeal (56.2-7).

##### 56.2-5 LeftIdealByGenerators
 ‣ LeftIdealByGenerators( R, gens ) ( operation )

LeftIdealByGenerators returns the ring that is generated by the elements of the collection gens under addition, multiplication, and multiplication with elements of the ring R from the left.

R can be accessed by LeftActingRingOfIdeal (56.2-10), gens can be accessed by GeneratorsOfLeftIdeal (56.2-8).

##### 56.2-6 RightIdealByGenerators
 ‣ RightIdealByGenerators( R, gens ) ( operation )

RightIdealByGenerators returns the ring that is generated by the elements of the collection gens under addition, multiplication, and multiplication with elements of the ring R from the right.

R can be accessed by RightActingRingOfIdeal (56.2-10), gens can be accessed by GeneratorsOfRightIdeal (56.2-9).

##### 56.2-7 GeneratorsOfTwoSidedIdeal
 ‣ GeneratorsOfTwoSidedIdeal( I ) ( attribute )
 ‣ GeneratorsOfIdeal( I ) ( attribute )

is a list of generators for the ideal I, with respect to the action of the rings that are stored as the values of LeftActingRingOfIdeal (56.2-10) and RightActingRingOfIdeal (56.2-10), from the left and from the right, respectively.

gap> A:= FullMatrixAlgebra( Rationals, 3 );;
gap> I:= Ideal( A, [ One( A ) ] );;
gap> GeneratorsOfIdeal( I );
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]


##### 56.2-8 GeneratorsOfLeftIdeal
 ‣ GeneratorsOfLeftIdeal( I ) ( attribute )

is a list of generators for the left ideal I, with respect to the action from the left of the ring that is stored as the value of LeftActingRingOfIdeal (56.2-10).

##### 56.2-9 GeneratorsOfRightIdeal
 ‣ GeneratorsOfRightIdeal( I ) ( attribute )

is a list of generators for the right ideal I, with respect to the action from the right of the ring that is stored as the value of RightActingRingOfIdeal (56.2-10).

##### 56.2-10 LeftActingRingOfIdeal
 ‣ LeftActingRingOfIdeal( I ) ( attribute )
 ‣ RightActingRingOfIdeal( I ) ( attribute )

returns the left (resp. right) acting ring of an ideal I.

##### 56.2-11 AsLeftIdeal
 ‣ AsLeftIdeal( R, S ) ( operation )
 ‣ AsRightIdeal( R, S ) ( operation )
 ‣ AsTwoSidedIdeal( R, S ) ( operation )

Let S be a subring of the ring R.

If S is a left ideal in R then AsLeftIdeal returns this left ideal, otherwise fail is returned.

If S is a right ideal in R then AsRightIdeal returns this right ideal, otherwise fail is returned.

If S is a two-sided ideal in R then AsTwoSidedIdeal returns this two-sided ideal, otherwise fail is returned.

gap> A:= FullMatrixAlgebra( Rationals, 3 );;
gap> B:= DirectSumOfAlgebras( A, A );
<algebra over Rationals, with 6 generators>
gap> C:= Subalgebra( B, Basis( B ){[1..9]} );
<algebra over Rationals, with 9 generators>
gap> I:= AsTwoSidedIdeal( B, C );
<two-sided ideal in <algebra of dimension 18 over Rationals>,
(9 generators)>


#### 56.3 Rings With One

##### 56.3-1 IsRingWithOne
 ‣ IsRingWithOne( R ) ( filter )

A ring-with-one in GAP is a ring (see IsRing (56.1-1)) that is also a magma-with-one (see IsMagmaWithOne (35.1-2)).

Note that the identity and the zero of a ring-with-one need not be distinct. This means that a ring that consists only of its zero element can be regarded as a ring-with-one.

This is especially useful in the case of finitely presented rings, in the sense that each factor of a ring-with-one is again a ring-with-one.

##### 56.3-2 RingWithOne
 ‣ RingWithOne( r, s, ... ) ( function )
 ‣ RingWithOne( coll ) ( function )

In the first form RingWithOne returns the smallest ring with one that contains all the elements r, s, ... In the second form RingWithOne returns the smallest ring with one that contains all the elements in the collection C. If any element is not an element of a ring or if the elements lie in no common ring an error is raised.

gap> RingWithOne( [ 4, 6 ] );
Integers


##### 56.3-3 RingWithOneByGenerators
 ‣ RingWithOneByGenerators( coll ) ( operation )

RingWithOneByGenerators returns the ring-with-one generated by the elements in the collection coll, i. e., the closure of coll under addition, multiplication, taking additive inverses, and taking the identity of an element.

##### 56.3-4 GeneratorsOfRingWithOne
 ‣ GeneratorsOfRingWithOne( R ) ( attribute )

GeneratorsOfRingWithOne returns a list of elements such that the ring R is the closure of these elements under addition, multiplication, taking additive inverses, and taking the identity element One( R ).

R itself need not be known to be a ring-with-one.

gap> R:= RingWithOne( [ 4, 6 ] );
Integers
gap> GeneratorsOfRingWithOne( R );
[ 1 ]


##### 56.3-5 SubringWithOne
 ‣ SubringWithOne( R, gens ) ( function )
 ‣ SubringWithOneNC( R, gens ) ( function )

returns the ring with one with parent R generated by the elements in gens. When the second form, SubringWithOneNC is used, it is not checked whether all elements in gens lie in R.

gap> R:= SubringWithOne( Integers, [ 4, 6 ] );
Integers
gap> Parent( R );
Integers


#### 56.4 Properties of Rings

##### 56.4-1 IsIntegralRing
 ‣ IsIntegralRing( R ) ( property )

A ring-with-one R is integral if it is commutative, contains no nontrivial zero divisors, and if its identity is distinct from its zero.

gap> IsIntegralRing( Integers );
true


##### 56.4-2 IsUniqueFactorizationRing
 ‣ IsUniqueFactorizationRing( R ) ( category )

A ring R is called a unique factorization ring if it is an integral ring (see IsIntegralRing (56.4-1)), and every nonzero element has a unique factorization into irreducible elements, i.e., a unique representation as product of irreducibles (see IsIrreducibleRingElement (56.5-7)). Unique in this context means unique up to permutations of the factors and up to multiplication of the factors by units (see Units (56.5-2)).

Mathematically, a field should therefore also be a unique factorization ring, since every nonzero element is a unit. In GAP, however, at least at present fields do not lie in the filter IsUniqueFactorizationRing, since operations such as Factors (56.5-9), Gcd (56.7-1), StandardAssociate (56.5-5) and so on do not apply to fields (the results would be trivial, and not especially useful) and methods which require their arguments to lie in IsUniqueFactorizationRing expect these operations to work.

(Note that we cannot install a subset maintained method for this filter since the factorization of an element needs not exist in a subring. As an example, consider the subring 4 ℕ + 1 of the ring 4 ℤ + 1; in the subring, the element 3 ⋅ 3 ⋅ 11 ⋅ 7 has the two factorizations 33 ⋅ 21 = 9 ⋅ 77, but in the large ring there is the unique factorization (-3) ⋅ (-3) ⋅ (-11) ⋅ (-7), and it is easy to see that every element in 4 ℤ + 1 has a unique factorization.)

gap> IsUniqueFactorizationRing( PolynomialRing( Rationals, 1 ) );
true


##### 56.4-3 IsLDistributive
 ‣ IsLDistributive( C ) ( property )

is true if the relation a * ( b + c ) = ( a * b ) + ( a * c ) holds for all elements a, b, c in the collection C, and false otherwise.

##### 56.4-4 IsRDistributive
 ‣ IsRDistributive( C ) ( property )

is true if the relation ( a + b ) * c = ( a * c ) + ( b * c ) holds for all elements a, b, c in the collection C, and false otherwise.

##### 56.4-5 IsDistributive
 ‣ IsDistributive( C ) ( property )

is true if the collection C is both left and right distributive (see IsLDistributive (56.4-3), IsRDistributive (56.4-4)), and false otherwise.

gap> IsDistributive( Integers );
true


##### 56.4-6 IsAnticommutative
 ‣ IsAnticommutative( R ) ( property )

is true if the relation a * b = - b * a holds for all elements a, b in the ring R, and false otherwise.

##### 56.4-7 IsZeroSquaredRing
 ‣ IsZeroSquaredRing( R ) ( property )

is true if a * a is the zero element of the ring R for all a in R, and false otherwise.

##### 56.4-8 IsJacobianRing
 ‣ IsJacobianRing( R ) ( property )

is true if the Jacobi identity holds in the ring R, and false otherwise. The Jacobi identity means that x * (y * z) + z * (x * y) + y * (z * x) is the zero element of R, for all elements x, y, z in R.

gap> L:= FullMatrixLieAlgebra( GF( 5 ), 7 );
<Lie algebra over GF(5), with 13 generators>
gap> IsJacobianRing( L );
true


#### 56.5 Units and Factorizations

##### 56.5-1 IsUnit
 ‣ IsUnit( [R, ]r ) ( operation )

IsUnit returns true if r is a unit in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)). If r is not a unit then false is returned.

An element r is called a unit in a ring R, if r has an inverse in R.

IsUnit may call Quotient (56.1-9).

##### 56.5-2 Units
 ‣ Units( R ) ( attribute )

Units returns the group of units of the ring R. This may either be returned as a list or as a group.

An element r is called a unit of a ring R if r has an inverse in R. It is easy to see that the set of units forms a multiplicative group.

gap> Units( GaussianIntegers );
[ -1, 1, -E(4), E(4) ]
gap> Units( GF( 16 ) );
<group of size 15 with 1 generators>


##### 56.5-3 IsAssociated
 ‣ IsAssociated( [R, ]r, s ) ( operation )

IsAssociated returns true if the two ring elements r and s are associated in the ring R, if given, and otherwise in their default ring (see DefaultRing (56.1-3)). If the two elements are not associated then false is returned.

Two elements r and s of a ring R are called associated if there is a unit u of R such that r u =s.

##### 56.5-4 Associates
 ‣ Associates( [R, ]r ) ( operation )

Associates returns the set of associates of r in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)).

Two elements r and s of a ring R are called associated if there is a unit u of R such that r u = s.

gap> Associates( Integers, 2 );
[ -2, 2 ]
gap> Associates( GaussianIntegers, 2 );
[ -2, 2, -2*E(4), 2*E(4) ]


##### 56.5-5 StandardAssociate
 ‣ StandardAssociate( [R, ]r ) ( operation )

StandardAssociate returns the standard associate of the ring element r in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)).

The standard associate of a ring element r of R is an associated element of r which is, in a ring dependent way, distinguished among the set of associates of r. For example, in the ring of integers the standard associate is the absolute value.

gap> x:= Indeterminate( Rationals, "x" );;
gap> StandardAssociate( -x^2-x+1 );
x^2+x-1


##### 56.5-6 StandardAssociateUnit
 ‣ StandardAssociateUnit( [R, ]r ) ( operation )

StandardAssociateUnit returns a unit in the ring R such that the ring element r times this unit equals the standard associate of r in R.

If R is not given, the default ring of r is used instead. (see DefaultRing (56.1-3)).

gap> y:= Indeterminate( Rationals, "y" );;
gap> r:= -y^2-y+1;
-y^2-y+1
gap> StandardAssociateUnit( r );
-1
gap> StandardAssociateUnit( r ) * r = StandardAssociate( r );
true


##### 56.5-7 IsIrreducibleRingElement
 ‣ IsIrreducibleRingElement( [R, ]r ) ( operation )

IsIrreducibleRingElement returns true if the ring element r is irreducible in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)). If r is not irreducible then false is returned.

An element r of a ring R is called irreducible if r is not a unit in R and if there is no nontrivial factorization of r in R, i.e., if there is no representation of r as product s t such that neither s nor t is a unit (see IsUnit (56.5-1)). Each prime element (see IsPrime (56.5-8)) is irreducible.

gap> IsIrreducibleRingElement( Integers, 2 );
true


##### 56.5-8 IsPrime
 ‣ IsPrime( [R, ]r ) ( operation )

IsPrime returns true if the ring element r is a prime in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)). If r is not a prime then false is returned.

An element r of a ring R is called prime if for each pair s and t such that r divides s t the element r divides either s or t. Note that there are rings where not every irreducible element (see IsIrreducibleRingElement (56.5-7)) is a prime.

##### 56.5-9 Factors
 ‣ Factors( [R, ]r ) ( operation )

Factors returns the factorization of the ring element r in the ring R, if given, and otherwise in in its default ring (see DefaultRing (56.1-3)). The factorization is returned as a list of primes (see IsPrime (56.5-8)). Each element in the list is a standard associate (see StandardAssociate (56.5-5)) except the first one, which is multiplied by a unit as necessary to have Product( Factors( R, r ) ) = r. This list is usually also sorted, thus smallest prime factors come first. If r is a unit or zero, Factors( R, r ) = [ r ].

gap> x:= Indeterminate( GF(2), "x" );;
gap> pol:= x^2+x+1;
x^2+x+Z(2)^0
gap> Factors( pol );
[ x^2+x+Z(2)^0 ]
gap> Factors( PolynomialRing( GF(4) ), pol );
[ x+Z(2^2), x+Z(2^2)^2 ]


 ‣ PadicValuation( r, p ) ( operation )

PadicValuation is the operation to compute the p-adic valuation of a ring element r.

#### 56.6 Euclidean Rings

##### 56.6-1 IsEuclideanRing
 ‣ IsEuclideanRing( R ) ( category )

A ring R is called a Euclidean ring if it is an integral ring and there exists a function δ, called the Euclidean degree, from R-{0_R} into a well-ordered set (such as the the nonnegative integers), such that for every pair r ∈ R and s ∈ R-{0_R} there exists an element q such that either r - q s = 0_R or δ(r - q s) < δ( s ). In GAP the Euclidean degree δ is implicitly built into a ring and cannot be changed. The existence of this division with remainder implies that the Euclidean algorithm can be applied to compute a greatest common divisor of two elements, which in turn implies that R is a unique factorization ring.

gap> IsEuclideanRing( GaussianIntegers );
true


##### 56.6-2 EuclideanDegree
 ‣ EuclideanDegree( [R, ]r ) ( operation )

EuclideanDegree returns the Euclidean degree of the ring element r in the ring R, if given, and otherwise in its default ring (see DefaultRing (56.1-3)).

The ring R must be a Euclidean ring (see IsEuclideanRing (56.6-1)).

gap> EuclideanDegree( GaussianIntegers, 3 );
9


##### 56.6-3 EuclideanQuotient
 ‣ EuclideanQuotient( [R, ]r, m ) ( operation )

EuclideanQuotient returns the Euclidean quotient of the ring elements r and m in the ring R, if given, and otherwise in their default ring (see DefaultRing (56.1-3)).

The ring R must be a Euclidean ring (see IsEuclideanRing (56.6-1)), otherwise an error is signalled.

gap> EuclideanQuotient( 8, 3 );
2


##### 56.6-4 EuclideanRemainder
 ‣ EuclideanRemainder( [R, ]r, m ) ( operation )

EuclideanRemainder returns the Euclidean remainder of the ring element r modulo the ring element m in the ring R, if given, and otherwise in their default ring (see DefaultRing (56.1-3)).

The ring R must be a Euclidean ring (see IsEuclideanRing (56.6-1)), otherwise an error is signalled.

gap> EuclideanRemainder( 8, 3 );
2


##### 56.6-5 QuotientRemainder
 ‣ QuotientRemainder( [R, ]r, m ) ( operation )

QuotientRemainder returns the Euclidean quotient and the Euclidean remainder of the ring elements r and m in the ring R, if given, and otherwise in their default ring (see DefaultRing (56.1-3)). The result is a pair of ring elements.

The ring R must be a Euclidean ring (see IsEuclideanRing (56.6-1)), otherwise an error is signalled.

gap> QuotientRemainder( GaussianIntegers, 8, 3 );
[ 3, -1 ]


#### 56.7 Gcd and Lcm

##### 56.7-1 Gcd
 ‣ Gcd( [R, ]r1, r2, ... ) ( function )
 ‣ Gcd( [R, ]list ) ( function )

Gcd returns the greatest common divisor of the ring elements r1, r2, ... resp. of the ring elements in the list list in the ring R, if given, and otherwise in their default ring, see DefaultRing (56.1-3).

Gcd returns the standard associate (see StandardAssociate (56.5-5)) of the greatest common divisors.

A divisor of an element r in the ring R is an element d∈ R such that r is a multiple of d. A common divisor of the elements r_1, r_2, ... in the ring R is an element d∈ R which is a divisor of each r_1, r_2, .... A greatest common divisor d in addition has the property that every other common divisor of r_1, r_2, ... is a divisor of d.

Note that this in particular implies the following: For the zero element z of R, we have Gcd( r, z ) = Gcd( z, r ) = StandardAssociate( r ) and Gcd( z, z ) = z.

gap> Gcd( Integers, [ 10, 15 ] );
5


##### 56.7-2 GcdOp
 ‣ GcdOp( [R, ]r, s ) ( operation )

GcdOp is the operation to compute the greatest common divisor of two ring elements r, s in the ring R or in their default ring.

##### 56.7-3 GcdRepresentation
 ‣ GcdRepresentation( [R, ]r1, r2, ... ) ( function )
 ‣ GcdRepresentation( [R, ]list ) ( function )

GcdRepresentation returns a representation of the greatest common divisor of the ring elements r1, r2, ... resp. of the ring elements in the list list in the Euclidean ring R, if given, and otherwise in their default ring, see DefaultRing (56.1-3).

A representation of the gcd g of the elements r_1, r_2, ... of a ring R is a list of ring elements s_1, s_2, ... of R, such that g = s_1 r_1 + s_2 r_2 + ⋯. Such representations do not exist in all rings, but they do exist in Euclidean rings (see IsEuclideanRing (56.6-1)), which can be shown using the Euclidean algorithm, which in fact can compute those coefficients.

gap> a:= Indeterminate( Rationals, "a" );;
gap> GcdRepresentation( a^2+1, a^3+1 );
[ -1/2*a^2-1/2*a+1/2, 1/2*a+1/2 ]


Gcdex (14.3-5) provides similar functionality over the integers.

##### 56.7-4 GcdRepresentationOp
 ‣ GcdRepresentationOp( [R, ]r, s ) ( operation )

GcdRepresentationOp is the operation to compute the representation of the greatest common divisor of two ring elements r, s in the Euclidean ring R or in their default ring, respectively.

##### 56.7-5 ShowGcd
 ‣ ShowGcd( a, b ) ( function )

This function takes two elements a and b of an Euclidean ring and returns their greatest common divisor. It will print out the steps performed by the Euclidean algorithm, as well as the rearrangement of these steps to express the gcd as a ring combination of a and b.

gap> ShowGcd(192,42);
192=4*42 + 24
42=1*24 + 18
24=1*18 + 6
18=3*6 + 0
The Gcd is 6
= 1*24 -1*18
= -1*42 + 2*24
= 2*192 -9*42
6


##### 56.7-6 Lcm
 ‣ Lcm( [R, ]r1, r2, ... ) ( function )
 ‣ Lcm( [R, ]list ) ( function )

Lcm returns the least common multiple of the ring elements r1, r2, ... resp. of the ring elements in the list list in the ring R, if given, and otherwise in their default ring, see DefaultRing (56.1-3).

Lcm returns the standard associate (see StandardAssociate (56.5-5)) of the least common multiples.

A least common multiple of the elements r_1, r_2, ... of the ring R is an element m that is a multiple of r_1, r_2, ..., and every other multiple of these elements is a multiple of m.

Note that this in particular implies the following: For the zero element z of R, we have Lcm( r, z ) = Lcm( z, r ) = StandardAssociate( r ) and Lcm( z, z ) = z.

##### 56.7-7 LcmOp
 ‣ LcmOp( [R, ]r, s ) ( operation )

LcmOp is the operation to compute the least common multiple of two ring elements r, s in the ring R or in their default ring, respectively.

The default methods for this uses the equality lcm( m, n ) = m*n / gcd( m, n ) (see GcdOp (56.7-2)).

##### 56.7-8 QuotientMod
 ‣ QuotientMod( [R, ]r, s, m ) ( operation )

QuotientMod returns a quotient of the ring elements r and s modulo the ring element m in the ring R, if given, and otherwise in their default ring, see DefaultRing (56.1-3).

R must be a Euclidean ring (see IsEuclideanRing (56.6-1)) so that EuclideanRemainder (56.6-4) can be applied. If no modular quotient exists, fail is returned.

A quotient q of r and s modulo m is an element of R such that q s = r modulo m, i.e., such that q s - r is divisible by m in R and that q is either zero (if r is divisible by m) or the Euclidean degree of q is strictly smaller than the Euclidean degree of m.

gap> QuotientMod( 7, 2, 3 );
2


##### 56.7-9 PowerMod
 ‣ PowerMod( [R, ]r, e, m ) ( operation )

PowerMod returns the e-th power of the ring element r modulo the ring element m in the ring R, if given, and otherwise in their default ring, see DefaultRing (56.1-3). e must be an integer.

R must be a Euclidean ring (see IsEuclideanRing (56.6-1)) so that EuclideanRemainder (56.6-4) can be applied to its elements.

If e is positive the result is r^e modulo m. If e is negative then PowerMod first tries to find the inverse of r modulo m, i.e., i such that i r = 1 modulo m. If the inverse does not exist an error is signalled. If the inverse does exist PowerMod returns PowerMod( R, i, -e, m ).

PowerMod reduces the intermediate values modulo m, improving performance drastically when e is large and m small.

gap> PowerMod( 12, 100000, 7 );
2


##### 56.7-10 InterpolatedPolynomial
 ‣ InterpolatedPolynomial( R, x, y ) ( operation )

InterpolatedPolynomial returns, for given lists x, y of elements in a ring R of the same length n, say, the unique polynomial of degree less than n which has value y[i] at x[i], for all i ∈ { 1, ..., n }. Note that the elements in x must be distinct.

gap> InterpolatedPolynomial( Integers, [ 1, 2, 3 ], [ 5, 7, 0 ] );
-9/2*x^2+31/2*x-6


#### 56.8 Homomorphisms of Rings

A ring homomorphism is a mapping between two rings that respects addition and multiplication.

Currently GAP supports ring homomorphisms between finite rings (using straightforward methods) and ring homomorphisms with additional structures, where source and range are in fact algebras and where also the linear structure is respected, see 62.10.

##### 56.8-1 RingGeneralMappingByImages
 ‣ RingGeneralMappingByImages( R, S, gens, imgs ) ( operation )

is a general mapping from the ring A to the ring S. This general mapping is defined by mapping the entries in the list gens (elements of R) to the entries in the list imgs (elements of S), and taking the additive and multiplicative closure.

gens need not generate R as a ring, and if the specification does not define an additive and multiplicative mapping then the result will be multivalued. Hence, in general it is not a mapping.

##### 56.8-2 RingHomomorphismByImages
 ‣ RingHomomorphismByImages( R, S, gens, imgs ) ( function )

RingHomomorphismByImages returns the ring homomorphism with source R and range S that is defined by mapping the list gens of generators of R to the list imgs of images in S.

If gens does not generate R or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then fail is returned.

One can avoid the checks by calling RingHomomorphismByImagesNC (56.8-3), and one can construct multi-valued mappings with RingGeneralMappingByImages (56.8-1).

##### 56.8-3 RingHomomorphismByImagesNC
 ‣ RingHomomorphismByImagesNC( R, S, gens, imgs ) ( operation )

RingHomomorphismByImagesNC is the operation that is called by the function RingHomomorphismByImages (56.8-2). Its methods may assume that gens generates R as a ring and that the mapping of gens to imgs defines a ring homomorphism. Results are unpredictable if these conditions do not hold.

For creating a possibly multi-valued mapping from R to S that respects addition and multiplication, RingGeneralMappingByImages (56.8-1) can be used.

##### 56.8-4 NaturalHomomorphismByIdeal
 ‣ NaturalHomomorphismByIdeal( R, I ) ( operation )

is the homomorphism of rings provided by the natural projection map of R onto the quotient ring R/I. This map can be used to take pre-images in the original ring from elements in the quotient.

#### 56.9 Small Rings

GAP contains a library of small (order up to 15) rings.

##### 56.9-1 SmallRing
 ‣ SmallRing( s, n ) ( function )

returns the n-th ring of order s from a library of rings of small order (up to isomorphism).

gap> R:=SmallRing(8,37);
<ring with 3 generators>
gap> ShowMultiplicationTable(R);
*     | 0*a   c     b     b+c   a     a+c   a+b   a+b+c
------+------------------------------------------------
0*a   | 0*a   0*a   0*a   0*a   0*a   0*a   0*a   0*a
c     | 0*a   0*a   0*a   0*a   0*a   0*a   0*a   0*a
b     | 0*a   0*a   0*a   0*a   b     b     b     b
b+c   | 0*a   0*a   0*a   0*a   b     b     b     b
a     | 0*a   c     b     b+c   a+b   a+b+c a     a+c
a+c   | 0*a   c     b     b+c   a+b   a+b+c a     a+c
a+b   | 0*a   c     b     b+c   a     a+c   a+b   a+b+c
a+b+c | 0*a   c     b     b+c   a     a+c   a+b   a+b+c


##### 56.9-2 NumberSmallRings
 ‣ NumberSmallRings( s ) ( function )

returns the number of (nonisomorphic) rings of order s stored in the library of small rings.

gap> List([1..15],NumberSmallRings);
[ 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4 ]


##### 56.9-3 Subrings
 ‣ Subrings( R ) ( attribute )

for a finite ring R this function returns a list of all subrings of R.

gap> Subrings(SmallRing(8,37));
[ <ring with 1 generators>, <ring with 1 generators>,
<ring with 1 generators>, <ring with 1 generators>,
<ring with 1 generators>, <ring with 1 generators>,
<ring with 2 generators>, <ring with 2 generators>,
<ring with 2 generators>, <ring with 2 generators>,
<ring with 3 generators> ]


##### 56.9-4 Ideals
 ‣ Ideals( R ) ( attribute )

for a finite ring R this function returns a list of all ideals of R.

gap> Ideals(SmallRing(8,37));
[ <ring with 1 generators>, <ring with 1 generators>,
<ring with 1 generators>, <ring with 2 generators>,
<ring with 3 generators> ]


##### 56.9-5 DirectSum
 ‣ DirectSum( R{, S} ) ( function )
 ‣ DirectSumOp( list, expl ) ( operation )

These functions construct the direct sum of the rings given as arguments. DirectSum takes an arbitrary positive number of arguments and calls the operation DirectSumOp, which takes exactly two arguments, namely a nonempty list of rings and one of these rings. (This somewhat strange syntax allows the method selection to choose a reasonable method for special cases.)

gap> DirectSum(SmallRing(5,1),SmallRing(5,1));
<ring with 2 generators>


##### 56.9-6 RingByStructureConstants
 ‣ RingByStructureConstants( moduli, sctable[, nameinfo] ) ( function )

returns a ring R whose additive group is described by the list moduli, with multiplication defined by the structure constants table sctable. The optional argument nameinfo can be used to prescribe names for the elements of the canonical generators of R; it can be either a string name (then name1, name2 etc. are chosen) or a list of strings which are then chosen.

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