Eureka Math: A Story of Ratios Contributors Michael Allwood, Curriculum Writer Tiah Alphonso, Program Manager—Curriculum Production Catriona Anderson, Program Manager—Implemen

“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).

{0} ⊂ T is a sub-module. It is called the to sion submodule of M. Proof. Because a domain has no zero-divisors, we can conclude that: a1t1 = 0 and a2t2 = 0 implies a1a2(t1 + t2) = 0 and a1, a2 6= 0 …

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.

The left R-module ( defined in Theorem 1.18) having the quotient group M/N for its underlying abelian group is called the quotient module ( or a factor module) of M modulo N and is denoted by R(M/N) or …

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 …

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 space …