Suppose is a prime number and is a commutative ring with unity of characteristic 0 in which is not a unit. Assume that and are -primary abelian groups such that the respective group algebras and are -isomorphic. Under certain restrictions on the ideal structure of , it is shown that and are isomorphic.
In this note we study sets of normal generators of finitely presented residually -finite groups. We show that if an infinite, finitely presented, residually -finite group is normally generated by with order , then where denotes the first -Betti number of . We also show that any -generated group with must have girth greater than or equal .
We show in an additive inverse regular semiring with as the set of all multiplicative idempotents and as the set of all additive idempotents, the following conditions are equivalent: (i) For all , implies . (ii) is orthodox. (iii) is a semilattice of groups. This result generalizes the corresponding result of regular ring.
We provide a characterization of artin algebras without chains of nonzero homomorphisms between indecomposable finitely generated modules starting with an injective module and ending with a projective module.