53.4 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).

(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

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