Question

What is an irreducible module?

(a) A cyclic module in a ring with any non-zero element as its generator

(b) A cyclic module in a ring with any positive integer as its generator

(c) An acyclic module in a ring with rational elements as its generator

(d) A linearly independent module in a semigroup with a set of real numbers

This question was addressed to me in final exam.

The above asked question is from Cyclic Groups in division Groups of Discrete Mathematics

Answer 1

The answer is (A) A cyclic module in a ring with any non-zero element as its generator

A nonzero R-module M is irreducible if and only if M is a cyclic module with any nonzero element as its generator. Suppose that M is an irreducible module. Let a∈M be any nonzero element and consider the submodule (a) generated by the element a. Since a is a nonzero element, the submodule (a) is non-zero. Since M is irreducible, this implies that M=(a). Hence M is a cyclic module generated by a. Since a is any nonzero element, module M is a cyclic module with any nonzero element as its generator.