Abelian ring

Abelian ring

[ə′bēl·yən ′riŋ]
(mathematics)
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0 x y z 0 0 0 0 0 x 0 0 0 0 y 0 0 y y z 0 0 y y It is an abelian ring. With respect to these two tables, {0, x} and {0, y} are two ideals of R.
0 x y z 0 0 0 0 0 x 0 0 0 0 y 0 0 0 0 z 0 x y x It is not an abelian ring. With respect to these two tables, {0, x} is an ideal of R but not completely prime ideal.
Note that M is an abelian module if and only if S is an abelian ring. In [9, Proposition 7], it is shown that the class of abelian generalized right principally projective rings is closed under direct sums.
However they require [SIGMA] to be an abelian ring and the CA to be a ring homomorphism, which is again essentially the case d = 1.
According to the Cayley-Hamilton theorem, which we can apply in our case because T is an endomorphism of a finite-dimensional free module over an abelian ring (see Theorem 3.1.