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N-COMMUTATIVE RINGS M(PartT) by IX v> Richard Brauer E. Weiss T-f,r. y I { e HARVARD UNIVERSITY NON-COMMUTATIVE RINGS (Part I) by Richard Brauer E. Weiss HARVARD UNIVERSITY Cambridge, Massachusetts Chapter I DEFINITION OF RINGS AND ALGEBRAS. EXAMPLES. In this chapter, we are concerned with the definition of rings and algebras. We then give a number of illustrations which will be useful and instructive later. 1. Definition of a ring. First properties. We consider a set R for which an addition and a multipli¬ cation are defined within R . This is to say that two functions s(x,y) and p(x,y) of two arguments x,y£R have been selected, both with values in R . We write x+y for s(x,y) and xy for p(x,y) . DEFINITION 1. A set R for which an addition and a multiplica¬ tion are defined in R is a ring, if (I) The elements of R form an abelian group under addition. (II) The multiplication is associative: (ab)c = a(bc) for arbitrary a, b, c 6 R . (III) Both distributive laws hold: a(b+c) = ab + ac , (b+c)a = ba + ca for arbitrary a, b, c in R . As a consequence of (I), we note that for given a, b € R the equation a+x=b has a unique solution x£R which is denoted by x=b-a . For a £ R , the element a-a is independent of a and is called the zero-element 0 of R . Then b+0=b for any b G R . As usual, we write -a for 0-a . Then b-a=b+(-a) , It follows from the associati- 2 - - vity of the addition that, for any positive integer n , we can form sums of n elements of R without using parentheses. In particular, we write na for a sum of n equal terms a € R . Similarly, because of (II), we can form products of n elements of R without using parentheses. The product of n equal factors a is of course denoted by a11 . Taking b=c=0 in (III), we see that a0=0, 0a=0 for every a £ R . It then follows from (III) for c=-b, b € R, that a(-b)= -ab, (-b)a=-ba . The ordinary rules for brackets hold. However, since the commutativity of multiplication was not assumed, we have to be careful about the order of the factors. A ring is said to be commutative, if ab=ba for all a, b £ R . Let R be a ring which does not only consist of the single element 0 . It may happen that there exists an element e in R such that ea=ae = a for all a € R . Then e ^0 and further e is uniquely deter¬ mined. We call e the unit element of R and often denote it by 1 . If R is a ring with a unit element 1 , an element a of R is said to be regular, if there exists an element a’ of R such that aa,=a,a=l . Then a' is uniquely determined by a . We usually write a * for a and call a * the inverse of a . DEFINITION 2. A skew field is a ring R with unit element such that every element a^=0 has an inverse. Alternately, we may say that the ring R is a skew field, iff the elements different from 0 form a group under multiplication. A field is a commutative skew field. Before we come to the discussion of examples, it should be mentioned that, as in all abstract theories, we are not interested in what the elements of R "are" but only how they "behave" under addition and multiplication. This leads to the idea of isomorphism. DEFINITION 3. An isomorphism of a ring R onto a ring R* is -3- a one-to-one mapping x —> x* of R onto R* such that (x+y)# = xsH-y*, (xy)* = x*y* for all x, y £ R . The rings R and R* are isomorphic, in symbols RiStfR#, if isomorphisms of R onto R* exist. If we have an isomorphism of R onto R* and if we agree to use the same letter for an element x of R and the corresponding element of R* , then any equation based on the ring properties in R is true in R* and vice versa. In a way, we may say that R and R* differ "in name only" or that the rings R and R* are "not essentially different". We cannot distinguish between them in an intrinsic manner using only ring properties. It will be clear that an isomorphism of R onto R* maps the zero-element of R on that of R* , the unit element of R (if it exists) on the unit element of R* , etc. We have (x-y)* = x*-y* for x, y £ R . 2. Rings of endomorphisms of abelian groups. We discuss a special class of rings which will be of fundamental importance. Let G be an abelian group in which the group operation is written as addition. Consider first the set M of all mappings 0 of G into G . Such mapping is given, if for every g£ G , the element is known on which g is mapped. We denote this image by g© or by 0(g) . Of course, the notion of a mapping is the same as that of a function defined on G with values in G . Two mappings 0 and 0’fe M are equal, iff g0 = g0' for all g € G . The sum of two elements 0^, 0^ of M is defined by the formula (1) g(©1 + ©2) = g©1 + g©2 (for a11 g 6 G). Here the element on the right makes sense, since g0 , g0 £ G , and G 1 w is an additive group. In other words, 0 +0 is the mapping which maps X u the element g € G on the sum of its images g0 , g0 for the two X M -4- mappings 9^, 9^ , This is the usual definition of the sum of two functions. It will be clear from this that M itself is an abelian group under addition. The zero-element is the zero-mapping □ of G which maps every gC G on the zero-element 0 of G; (2) gD =0 (for all g € G). If 9 £ M , the mapping -9 is given by (3) g(-0) = -(g©) . Here, the element -(g9) is the negative of the element g9 of the additive group G . We next define the product of two transformations 61 * Q2€ M by the formula (4) g(9192) = (g©1)©2 (for g€G). Thus, *s t^ie maPP^ng obtained by performing 9^ first which maps g —and following it up by 9^ which takes g9^—> (g9^)9^ • It is seen at once that this multiplication is associative. Moreover, if 9 , 9^ , 9^ are three elements of M and if (T ^ = 9(9^+9^) and = 99^+99^ , it follows from the definitions of addition and multiplication of mappings that both and map g —> (g9)9^ + (g9)9^ • Hence O ^ , i. e., we have the left distributive law 9(9^9^ = 99x + 992 . However, M is not a ring, since the right distributive law is not true in general. In order to obtain a ring, we replace M by the subset E consisting of the endomorphisms of G . An endomorphism 9 of G is simply a homomorphic mapping of G into G , i.e.. a mapping of G into G such