Bmodule
A Bmodule, or B-module, is a module over a ring B. This means M is an abelian group equipped with an action B × M → M that satisfies compatibility with ring operations: for all b1, b2 in B and m in M, (b1 b2)·m = b1·(b2·m), (b1 + b2)·m = b1·m + b2·m, and b·(m1 + m2) = b·m1 + b·m2, with the unity condition 1·m = m if B has a multiplicative identity. The term left B-module specifies that the action uses elements of B on the left; if B is noncommutative, a right B-module uses a right action m·b with analogous axioms.
Submodules and quotients are central concepts. A submodule N of M is a subgroup closed under the
Typical examples clarify the idea. If B = Z, a B-module is simply an abelian group. If B
Further topics include finitely generated modules, subclasses such as projective, injective, and free modules, and constructions