54 Rings

A ring in GAP is an additive group (see IsAdditiveGroup) that is also a magma (see IsMagma), such that addition + and multiplication * are distributive.

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

In the first form Ring returns the smallest ring that contains all the elements r, s... etc. 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 (see 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> Ring( 2, E(4) );
<ring with 2 generators>

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 the ring functions like Quotient, IsPrime, Factors, or Gcd if no explicit ring is given.

Ring (see Ring) 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

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>

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

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 ]

If the elements in the collection C form a ring then AsRing returns this ring, otherwise fail is returned.

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 2 generators>
gap> Parent( S );
Integers

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>

In the first form Quotient returns the quotient of the two ring elements r and s in the ring R. In the second form Quotient returns the quotient of the two ring elements r and s in their default ring. 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

54.2 Ideals in 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 IsIdealInParent can be used to store whether a ring is an ideal in its parent.

(Whenever the term Ideal occurs without 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, GeneratorsOfRightIdeal, and GeneratorsOfTwoSidedIdeal. The acting rings can be accessed as LeftActingRingOfIdeal and RightActingRingOfIdeal, 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 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 IsIdeal is of course independent of the presence of the attribute values.

Ideals are constructed with LeftIdeal, RightIdeal, TwoSidedIdeal. 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.

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 or RightActingRingOfIdeal (or both) of I.

If R is a left F-module then also I is a left F-module, in particular the LeftActingDomain (see LeftActingDomain) 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)>

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

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

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 or RightActingRingOfIdeal, gens can be accessed by GeneratorsOfTwoSidedIdeal.

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, gens can be accessed by GeneratorsOfLeftIdeal.

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, gens can be accessed by GeneratorsOfRightIdeal.

is a list of generators for the bi-ideal I, with respect to the action of LeftActingRingOfIdeal( I ) from the left and the action of RightActingRingOfIdeal( I )from the right.

Note that LeftActingRingOfIdeal(I) and RightActingRingOfIdeal(I) coincide if I is a two-sided ideal.

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

is a list of generators for the left ideal I, with respect to the action of LeftActingRingOfIdeal( I ) from the left.

is a list of generators for the right ideal I, with respect to the action of RightActingRingOfIdeal( I ) from the right.

Let S be a subring of 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)>

54.3 Rings With One

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

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.

In the first form RingWithOne returns the smallest ring with one that contains all the elements r, s... etc. 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 ] );
<ring-with-one, with 2 generators>

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.

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 ] );
<ring-with-one, with 2 generators>
gap> GeneratorsOfRingWithOne( R );
[ 4, 6 ]

returns the ring with one 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:= SubringWithOne( Integers, [ 4, 6 ] );
<ring-with-one, with 2 generators>
gap> Parent( R );
Integers

54.4 Properties of Rings

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

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

Mathematically, a field should therefore also be a unique factorization ring, since every element is a unit. In GAP, however, at least at present fields do not lie in the filter IsUniqueFactorizationRing (see IsUniqueFactorizationRing), since Operations such as Factors, Gcd, StandardAssociate 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 category 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

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.

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.

is true if the collection C is both left and right distributive, and false otherwise.

gap> IsDistributive( Integers );
true

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

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

is true if the Jacobi identity holds in 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

54.5 Units and Factorizations

In the first form IsUnit returns true if r is a unit in the ring R. In the second form IsUnit returns true if the ring element r is a unit in its default ring (see DefaultRing).

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

IsUnit may call Quotient.

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 with 1 generators>

In the first form IsAssociated returns true if the two ring elements r and s are associated in the ring R and false otherwise. In the second form IsAssociated returns true if the two ring elements r and s are associated in their default ring (see DefaultRing) and false otherwise.

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.

In the first form Associates returns the set of associates of r in the ring R. In the second form Associates returns the set of associates of the ring element r in its default ring (see DefaultRing).

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) ]

In the first form StandardAssociate returns the standard associate of the ring element r in the ring R. In the second form StandardAssociate returns the standard associate of the ring element r in its default ring (see DefaultRing).

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 );
-1+x+x^2

In the first form IsIrreducibleRingElement returns true if the ring element r is irreducible in the ring R and false otherwise. In the second form IsIrreducibleRingElement returns true if the ring element r is irreducible in its default ring (see DefaultRing) and false otherwise.

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). Each prime element (see IsPrime) is irreducible.

gap> IsIrreducibleRingElement( Integers, 2 );
true

In the first form IsPrime returns true if the ring element r is a prime in the ring R and false otherwise. In the second form IsPrime returns true if the ring element r is a prime in its default ring (see DefaultRing) and false otherwise.

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) is a prime.

In the first form Factors returns the factorization of the ring element r in the ring R. In the second form Factors returns the factorization of the ring element r in its default ring (see DefaultRing). The factorization is returned as a list of primes (see IsPrime). Each element in the list is a standard associate (see StandardAssociate) 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( Rationals, "x" );;
gap> Factors( x^3+3*x^2+3*x+1 );
[ 1+x, 1+x, 1+x ]

54.6 Euclidean Rings

A ring R is called a Euclidean ring if it is an integral ring and there exists a function d, called the Euclidean degree, from R-{0_R} to the nonnegative integers, such that for every pair r Î R and s Î R-{0_R} there exists an element q such that either r - q s = 0_R or d(r - q s) < d( s ). In GAP the Euclidean degree d is implicitly built into an 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

In the first form EuclideanDegree returns the Euclidean degree of the ring element in the ring R. In the second form EuclideanDegree returns the Euclidean degree of the ring element r in its default ring. R must of course be a Euclidean ring (see IsEuclideanRing).

gap> EuclideanDegree( GaussianIntegers, 3 );
9

In the first form EuclideanQuotient returns the Euclidean quotient of the ring elements r and m in the ring R. In the second form EuclideanQuotient returns the Euclidean quotient of the ring elements r and m in their default ring. The ring R must be a Euclidean ring (see IsEuclideanRing) otherwise an error is signalled.

gap> EuclideanQuotient( 8, 3 );
2

In the first form EuclideanRemainder returns the remainder of the ring element r modulo the ring element m in the ring R. In the second form EuclideanRemainder returns the remainder of the ring element r modulo the ring element m in their default ring. The ring R must be a Euclidean ring (see IsEuclideanRing) otherwise an error is signalled.

gap> EuclideanRemainder( 8, 3 );
2

In the first form QuotientRemainder returns the Euclidean quotient and the Euclidean remainder of the ring elements r and m in the ring R. In the second form QuotientRemainder returns the Euclidean quotient and the Euclidean remainder of the ring elements r and m in their default ring as pair of ring elements. The ring R must be a Euclidean ring (see IsEuclideanRing) otherwise an error is signalled.

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

54.7 Gcd and Lcm

In the first two forms 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. In the second two forms Gcd returns the greatest common divisor of the ring elements r1, r2, ... resp. of the ring elements in the list list in their default ring (see DefaultRing). R must be a Euclidean ring (see IsEuclideanRing) so that QuotientRemainder (see QuotientRemainder) can be applied to its elements. Gcd returns the standard associate (see StandardAssociate) of the greatest common divisors.

A greatest common divisor of the elements r₁, r₂, ¼ of the ring R is an element of largest Euclidean degree (see EuclideanDegree) that is a divisor of r₁, r₂, ¼ .

We define Gcd( r, 0R ) = Gcd( 0R , r ) = StandardAssociate( r ) and Gcd( 0R , 0R ) = 0R .

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

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.

In the first two forms GcdRepresentation returns the representation of the greatest common divisor of the ring elements r1, r2, ... resp. of the ring elements in the list list in the ring R. In the second two forms GcdRepresentation returns the representation of the greatest common divisor of the ring elements r1, r2, ... resp. of the ring elements in the list list in their default ring (see DefaultRing). R must be a Euclidean ring (see IsEuclideanRing) so that Gcd (see Gcd) can be applied to its elements.

The representation of the gcd g of the elements r₁, r₂, ¼ of a ring R is a list of ring elements s₁, s₂, ¼ of R, such that g = s₁ r₁ + s₂ r₂ + ¼. That this representation exists can be shown using the Euclidean algorithm, which in fact can compute those coefficients.

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

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

In the first two forms 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. In the second two forms Lcm returns the least common multiple of the ring elements r1, r2, ... resp. of the ring elements in the list list in their default ring (see DefaultRing).

R must be a Euclidean ring (see IsEuclideanRing) so that Gcd (see Gcd) can be applied to its elements. Lcm returns the standard associate (see StandardAssociate) of the least common multiples.

A least common multiple of the elements r₁, r₂, ¼ of the ring R is an element of smallest Euclidean degree (see EuclideanDegree) that is a multiple of r₁, r₂, ¼ .

We define Lcm( r, 0R ) = Lcm( 0R , r ) = StandardAssociate( r ) and Lcm( 0R , 0R ) = 0R .

Lcm uses the equality lcm(m,n) = m*n / gcd(m,n) (see Gcd).

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.

In the first form QuotientMod returns the quotient of the ring elements r and s modulo the ring element m in the ring R. In the second form QuotientMod returns the quotient of the ring elements r and s modulo the ring element m in their default ring (see DefaultRing). R must be a Euclidean ring (see IsEuclideanRing) so that EuclideanRemainder (see EuclideanRemainder) can be applied. If the modular quotient does not exist, fail is returned.

The 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 0 (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

In the first form PowerMod returns the e-th power of the ring element r modulo the ring element m in the ring R. In the second form PowerMod returns the e-th power of the ring element r modulo the ring element m in their default ring (see DefaultRing). e must be an integer. R must be a Euclidean ring (see IsEuclideanRing) so that EuclideanRemainder (see EuclideanRemainder) 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

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 Î {1,¼,n}. Note that the elements in x must be distinct.

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

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GAP 4 manual
May 2002