S the inclusion mapping.

A matrix A" such that A" A = I; for any matrix X with n rows, AX = 0 implies X = 0; for any matrix Y with n columns, YA = 0 implies Y = o. 1, Chapter 5. We remark that when the left and right inverse both exist, they must be equal, for we have A' = lA' = A"AA' = A"I = A". Of course these conditions are no longer equivalent for a rectangular matrix. Generally, any matrix A satisfying (d) is called right regular; if A does not satisfy (d) and is not 0, it is called a left zero-divisor. Right zero-divisors and left regular matrices are defined similarly, using (e); further, regular means "left and right regular" (in agreement with the earlier definition) and a zero-divisor is an element or matrix which is a left or right zero-divisor.

Under what conditions can an abelian group be defined as a Z/(m)-module? 2. State the condition for two non-zero vectors in Z2 to be linearly dependent and give an example of two vectors that are linearly dependent, though neither is linearly dependent on the other. 3. Given a ring homomorphism I : R -+ S, show that every S-module can be considered as an R-module in a natural way. 4. Given two R-modules M, N and two homomorphisms I, 9 from M to N, show that the subset {x E Mlxl = xg} of M is a submodule.