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
Answer:A
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
Explanation: 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, the module M is a cyclic module with any nonzero element as its generator.