Algebra 819 - Homework 1 Samuel
نویسنده
چکیده
Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α + β)(m) = α(m) + β(m), (α ◦ β)(m) = α(β(m)). We must verify the four subring conditions from 3.2.2. (i) The additive identity of End(M) is z : M → M, m → 0M . Note that for any m,n ∈ M z(m + n) = 0M = 0M + 0M = z(m) + z(n) and for any r ∈ R, z(r ·m) = 0M = r · 0M = r · z(m). Thus, z ∈ EndR(M). (ii) Let α, β ∈ EndR(M). Then by the definitions of (End(M), +) and “R-linear” we get
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Algebra 819 - Homework 7
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