subsemimodules

A subsemimodule is a concept in abstract algebra, specifically within the study of semimodules. A semimodule is a generalization of a module, where the addition operation is only required to be associative and commutative, and the scalar multiplication distributes over the addition and is associative. A subsemimodule is a subset of a semimodule that is itself a semimodule under the same operations.

Formally, let S be a semiring and M be a left S-semimodule. A subset N of M is a subsemimodule of M if and only if it is non-empty and for any elements x, y in N and any scalar a in S, the following conditions hold: x + y is in N, and a * x is in N. This means that N is closed under the addition of elements from M and closed under scalar multiplication by elements from S.

The intersection of any collection of subsemimodules of a given semimodule is also a subsemimodule. The smallest subsemimodule containing a given subset of M is called the subsemimodule generated by that subset. If a subsemimodule N is equal to the entire semimodule M, then M is said to be a simple semimodule if its only subsemimodules are the zero submodule and itself.

The study of subsemimodules is crucial for understanding the structure of semimodules, analogous to how submodules are studied in module theory. Properties like freeness, projectivity, and injectivity can be examined in relation to subsemimodules.