“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 commutative).

This is definitely not true for R-modules if R is not a field— after all, any finite abelian group is a Z-module, but any free Z-module is the zero module or an infinite module.

nt modul q : N → N/f(M). Example. For the Z-module homomorphsm n : Z → Z, we have the modules: ker(n) = 0 ⊂ Z, im(n) = nZ ⊂ Z an q = co er(n) : Z → Z/nZ. Remark. This is a first isomorphism …

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) = aφ (m) . It …

Here we cover all the basic material on modules and vector spaces required for embarkation on advanced courses. Concerning the prerequisite algebraic background for this, we mention that any …

The resulting R-module M=N is called the quotient module of M with re-spect to the submodule N. The noether isomorphism theorems, which we have seen previously for groups and rings, then have …

A right module R is the same thing as a left Rop-module. Thus we may as well work with left modules, henceforth called modules, although there are few situations where it is convenient to work with right …