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By E. H. Connell
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Additional resources for Elements of Abstract and Linear Algebra
Show GT is a group. For each t ∈ T , let Gt = G. Note that GT is just another way Gt . Also note that if T = [0, 1] and G = R, the addition defined on of writing t∈T GT is just the usual addition of functions used in calculus. ) Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter 2, this chapter flows smoothly. This is because ideals are also normal subgroups and ring homomorphisms are also group homomorphisms.
If the elements of Rn are written as row vectors, Rn is identified with R1,n . , addition is a binary operation Rm,n × Rm,n → Rm,n . , the i, j term of the sum is the sum of the i, j terms. The following theorem is just an observation. Theorem Rm,n is an additive abelian group. Its “zero” is the matrix 0 = 0m,n all of whose terms are zero. Also −(ai,j ) = (−ai,j ). Furthermore, as additive groups, Rm,n ≈ Rmn . Scalar multiplication An element of R is called a scalar. A matrix may be “multiplied” on the right or left by a scalar.
Exercise Suppose R and S are rings. Note that R × 0 is not a subring of R × S because it does not contain (1R , 1S ). Show R × 0 is an ideal and (R × S/R × 0) ≈ S. ¯ ¯ ¯ ¯ Suppose I ⊂ R and J ⊂ S are ideals. Show I × J is an ideal of R × S and every ideal of R × S is of this form. Exercise Suppose R and S are commutative rings. Show T = R × S is not a domain. Let e = (1, 0) ∈ R × S and show e2 = e, (1 − e)2 = (1 − e), R × 0 = eT , and 0 × S = (1 − e)T . Exercise If T is any ring, an element e of T is called an idempotent provided e2 = e.
Elements of Abstract and Linear Algebra by E. H. Connell