By Arthur Wouk
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Extra info for A course of applied functional analysis
We say that the groupoid G can be isomorphically embedded in the groupoid G'9 if there exists an isomorphic map ping of the groupoid G onto some subgroupoid of the groupoid G'. This concept carries over, of course, to the case of rings. 6). e. matrices having the same element a along the principal diagonal, and zero everywhere outside this diagonal, form a subring in Rn9 isomorphic to the ring R. 6). In fact, functions which take one value a e R for all x from M form a subring in the ring of functions, which is isomorphic to the ring R.
There exist associative rings without divisors of zero which cannot be embedded in skew fields. [A. I. Mal'cev, Math. Ann. * 2. The purpose of this section is to prove that the necessary conditions mentioned above for the embedding of a semigroup in a group and of an associative ring in a skew field will also be sufficient in the commutative case. e. from ax — bx, where x e S, a, beG9 it always follows that a = b. Then the semigroup G can be isomorphically embedded in an Abelian semigroup G with identity such that every element of S possesses an inverse element in G.
In every ring R, any product in which at least one of the factors is equal to zero, is itself equal to zero. 0 = 0 (7) for all elements a from the ring R. In fact, if x is another arbitrary element of the ring R, then, in view of (2), a-0 = a(x — x) = ax — ax = 0. If a and b are any elements of an arbitrary ring R, then (-a)b = a(-b)= (-a)(-b) = ab. -ab, (8) (9) 29 GROUPS AND RINGS In fact, ab + ( - a)b = [a + (- a)]b = 0-6 = 0. In the same way we can verify the second half of equation (8). On the other hand, using (8), we obtain (-a)\-b) = -[a(-6)] = -(rab) = ab, which proves (9).
A course of applied functional analysis by Arthur Wouk