Algebra by L. E. Sigler (auth.)

By L. E. Sigler (auth.)

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Show that every nonzero element of R is multiplicatively invertible if and only if the equations ax+b=8 XC + d = 8 a, b, c, d E R, a # 8, b # 8 always have unique solutions (for x) in R. 8. Show that the commutativity of addition is derivable from the other statements in the definition of a unitary ring. 9. Let

We begin with the subring. Definition. S, a subset of the set R, is a sub ring of the ring ( R, +, ·, 0) if and only if (S, +, ·, 0) is a ring. 4 Subrings It is to be understood in this definition that + and ·,the binary operations on the subset S, are to have the same values on S that they have on the including set R. It is necessary, therefore, in order that + and · be binary operations on S, that x, yES imply x + y and xy belongs to S. We speak then of + and · as being closed on S. ExAMPLE. We denote the even integers by 27L = {2xlx E 7L}.

E) None of the four alternatives completes a satisfactory sentence. EXERCISES 1. Prove that the product of two rings is itself a ring. 2. Show that no product ring of non trivial rings can be an integral domain. 3. Give an example of a ring without a unity. 4. Does the ring <&>X, domain? +, n, 0) have nontrivial divisors of zero? Is it an integral 5. On the set 7L x 7L we define the following two operations: (s, t) + (u, v) = (s + u, t + v) + tv, sv + (s, t) D (u, v) = (su tu). Show that <7L x 7L, +, 0, (0, 0)) is a ring.

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