Exercise 35, § 1

Let A be a ring. a) The set $A^J$, endowed with the addition

and with the multiplication is a commutative ring, denoted by U(A). The identity element for addition is the sequence $0$ all of whose terms are zero; the identity element for multiplication is the sequence $1$ all of whose terms are zero except that of index 1, which is equal to $1_A$. The opposite of an element $a$ of U(A) is $I(a)$. b) Let $\rho : B \to A$ be a ring homomorphism. Then $U(\rho) : U(B) \to U(A)$ defined by $U(\rho)(b_n)_{n \in J} = (\rho(b_n))_{n \in J}$ is a ring homomorphism. c) The mapping $\Phi : U(A) \to A^J$ is a ring homomorphism. In other words $\Phi_n : U(A) \to A$ is a ring homomorphism for every $n \in J$. d) If the additive group of A has no $\mathbf{Z}$-torsion, the additive group of U(A) also has none. e) Let $q \in J$. If multiplication by $q$ is bijective in A, it is still bijective in U(A). (Use Exercise 32, c), p. 51.)