By Robert Wisbauer

This quantity offers a entire creation to module idea and the similar a part of ring thought, together with unique effects in addition to the latest paintings. it's a precious and stimulating learn for these new to the topic in addition to for researchers and serves as an quintessential reference quantity.

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**Example text**

R/I 2 is not semiprime. (c) ⇔ (d) results from the relationship between prime ideals in R and R/I. 18 Exercises. Verify for a ring R with unit : (1) The following assertions are equivalent: (a) R is a regular ring; (b) R is a direct sum of ideals I1 , . . , In , and every Ii is a regular ring. (2) The following assertions are equivalent: (a) R is fully idempotent; (b) for ideals I and left ideals A in R, we have A ∩ I ⊂ AI; (c) for ideals I and left ideals A in R with A ⊂ I, we have A ⊂ AI. (3) Every ideal A in the matrix ring R(n,n) is of the form I (n,n) for some ideal I ⊂ R.

Ke f is a submodule of M , Im f = (M )f is a submodule of N . For any submodule U of M , the factor group M/U = {m + U | m ∈ M } becomes an R-module by defining the operation of R on M/U r(m + U ) = rm + U, r ∈ R, m ∈ M, and M/U is called the factor (or quotient) module of M by U . The map pU : M → M/U, m → m + U, m ∈ M, 6 Elementary properties of modules 41 is surjective and a module homomorphism, called the canonical homomorphism (projection) of M onto M/U . It provides a bijection between the submodules of M containing U and the submodules of M/U .

Let M be a left module over R. e. rn ∈ N for all r ∈ R, n ∈ N . Then N is also an R-module by the operations induced from M : R × N → N, (r, n) → rn, r ∈ R, n ∈ N. M is called simple if M = 0 and it has no submodules except 0 and M . The submodules of R R (resp. R RR ) are just the left (resp. two-sided) ideals. For non-empty subsets N1 , N2 , N ⊂ M, A ⊂ R we define: N1 + N2 = {n1 + n2 | n1 ∈ N1 , n2 ∈ N2 } ⊂ M, k AN = { i=1 ai ni | ai ∈ A, ni ∈ N, k ∈ IN } ⊂ M. If N1 , N2 are submodules, then N1 + N2 is also a submodule of M .