“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 …

Before getting to our main subject matter, we open the text with another kind of structure, namely module, which plays a key role behind the scenes throughout algebra. In Chapter 1 we are …

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 …

If R is a ring and B is a submodule of an R-module A, then there is a one-to-one correspondence between the set of all submodules of A containing B and the set of all submodules of A/B, …

Show that M is indecomposable and not isomorphic to A, and that = A A as a left A-module. Thus the conclusion of the Krull-Schmidt theorem does not hold in this case (the theorem fails …

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) …

Introduction One of the most basic concepts in linear algebra is linear combinations: of vectors, of polynomials, of function. , and so on. For example, the polynomial 7 2T + 3T 2 is a linear …