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 action of B (b·n ∈ N for all b in B, n in N). If N ≤ M, then the quotient M/N inherits a natural B-module structure, and homomorphisms between B-modules preserve the action. The category of left B-modules, denoted Mod_B, has objects B-modules and morphisms that are B-linear maps.

Typical examples clarify the idea. If B = Z, a B-module is simply an abelian group. If B is a field F, a B-module is the same as a vector space over F. The ring B itself can be regarded as a left B-module via left multiplication, and finite direct sums of copies of B yield free B-modules.

Further topics include finitely generated modules, subclasses such as projective, injective, and free modules, and constructions like the tensor product, which combines a B-module with another module over B. Over commutative rings, many structural results parallel linear algebra; over noncommutative rings, left and right modules provide parallel theories. B-modules form a fundamental framework across algebra, representation theory, and algebraic geometry.