“Module” will always mean left module unless stated otherwise. Most of the time, there is no reason to switch the scalars from one side to the other (especially if the underlying ring is …

Modules are a generalization of the vector spaces of linear algebra in which the \scalars" are allowed to be from an arbitrary ring, rather than a ̄eld. This rather modest weakening of the …

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring. Modules also …

If S is a subring of R then any R-module can be considered as an S-module by restricting scalar multiplication to S M. For example, a complex vector space can be considered as a real vector …

All in all the approach chosen here leads to a clear refinement of the customary module theory and, for M = R, we obtain well-known results for the entire module category over a ring with unit.

As before, the basic examples are OX (a left D-module), X (a right D-module), DX (both a left and a right D-module). We see that the notion of a D-module on X is local.

Lemma 1.1. Let N be an A-module, then for φ ∈ HomB (M, N) there exists a unique ψ ∈ HomA (A ⊗B M, N) such that ψ j = φ. Proof. Clearly, ψ must satisfy the relation ψ (a ⊗ m) = aψ (1 ⊗ m) …