? ? ? 1. If R= Z, then condition (iii) of Definition 5.1 is superfluous (that is,

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1. If R= Z, then condition (iii) of Definition 5.1 is superfluous (that is, (i) and (ii) imply (iii)). 2. Let A and B be abelian groups. A/mA. (a) For each m > 0, AZ (b) Z. Z. Z., where c = (m,n). (c) Describe AB, when A and B are finitely generated. 3. If A is a torsion abelian group and Q the (additive) group of rationals, then (a) AQ 0. (b) Q&Q Q. Note: R is a ring and = z. Apply Definition 5.1. Let A be a right module and B a lefi module over a ring R. Let F be the free abelian group on the set AX B. Let K be the subgroup of F generated by all elements of the following forms (for all a,a' e A; b,b' e B; re R): (i) (a + a',b) (a,b) - (a,b); (ii) (a,b + b)(a,b) (a,b); (iii) (ar,b) (a,rb). The quotient group F/K is called the tensor product of A and B; it is denoted AOR B (or simply A BifR = Z). The coset (a,b) + K of the element (a,b) in F is denoted ab; the coset of (0,0) is denoted 0. 1. If R= Z, then condition (iii) of Definition 5.1 is superfluous (that is, (i) and (ii) imply (iii)). 2. Let A and B be abelian groups. A/mA. (a) For each m > 0, AZ (b) Z. Z. Z., where c = (m,n). (c) Describe AB, when A and B are finitely generated. 3. If A is a torsion abelian group and Q the (additive) group of rationals, then (a) AQ 0. (b) Q&Q Q. Note: R is a ring and = z. Apply Definition 5.1. Let A be a right module and B a lefi module over a ring R. Let F be the free abelian group on the set AX B. Let K be the subgroup of F generated by all elements of the following forms (for all a,a' e A; b,b' e B; re R): (i) (a + a',b) (a,b) - (a,b); (ii) (a,b + b)(a,b) (a,b); (iii) (ar,b) (a,rb). The quotient group F/K is called the tensor product of A and B; it is denoted AOR B (or simply A BifR = Z). The coset (a,b) + K of the element (a,b) in F is denoted ab; the coset of (0,0) is denoted 0.