By by James Hamblin Smith

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Even though the usage of these terms varies considerably from author to author, it is usually evident from context what is meant. Definition. Given a set S and a binary operation [3: S x S ~ S with neutral element v we say that an element x of S is invertible if and only if there exists an element y of S such that x[Jy = y[Jx = v. Moreover, if there is such an element y it is called the inverse of x. 33 2. Rings: Basic theory EXAMPLES. For 7L, +, 0 we see -2 to be the inverse of2. For IQ,·, 1 we see !

Given an arbitrary subset A of a ring (R, +, ·, 0) we ask the question whether or not there exist subrings of R which contain the given A as a subset. The answer is yes, there always exists at least one subring of R containing A, namely R itself. The following theorem proves a stronger result; there exists a smallest subring of R containing A. Theorem. Let A be any subset of a ring (R, +, ·, 0). Then there exists a smallest subring of R which contains the set A as a subset. 47 2. Rings: Basic theory PRooF.

Let (R, +, ·, fJ) be a ring. Then x(- y) = (-x)y = -(xy) for all X, yE R. PROOF. We have followed the usual custom of omitting the symbol for multiplication when no confusion can occur. x(- y) + xy = x[(- y) + y] = xfJ = fJ. Because + is commutative xy + x(- y) = (} also. We conclude x(- y) is the unique negative of xy which is written - (xy). This is to say, x(- y) = -(xy). In a symmetric manner we prove ( -x)y = -(xy). 0 Theorem. Let (R, +, ·, fJ) be a ring. Then (- x)(- y) = xy for all x, y e R.