The similarity between the properties of ideals in commutative rings and the properties of normal subgroups of groups

Abstract

Introduction. The object of this note is to point out a striking similarity between the properties of the set of normal subgroups of a group with ascending chain condition for normal subgroups and the properties of ideals of a commutative Noetherian ring. In particular it will be seen that a normal subgroup equal to its commutator subgroup is the intersection of pairwise complementary subgroups (analogue of pairwise relatively ideals). And any normal subgroup modulo which the whole group is semisimple, is the intersection of such subgroups which are irreducible. The theory is based on making the analogy in the following way: sum of ideals corresponds to the product of normal subgroups; product of ideals corresponds to the commutator of the normal subgroups; and the residual quotient of ideals has an analogue introduced here. The concepts prime ideal, irreducible ideal, and radical of an ideal have analogues for normal subgroups as will be pointed out. For the ideal theory one can consult Northcott [3 ] or van der Waerden [4].