The normal closure of the coproduct of rings over a division ring

Let R = R\UR 2 be the coproduct of ~-rings R\ and R2 with 1 over a division ring ~, RI "'~, R2 "'~, with at least one of the dimensions (R;: ~)r,(R;: ~)I' i = 1,2, greater than 2. If R\ and R2 are weakly I-finite (i.e., one-sided inverses are two-sided) then it is shown that every X-inner automorphism of R (in the sense of Kharchenko) is inner, unless R\, R2 satisfy one of the following conditions: (I) each R, is primary (i.e .• R, = ~ + T, T2 = 0). (II) one R; is primary and the other is 2-dimensional, (III) char. ~ = 2, one R, is not a domain. and one R; is 2-dimensional. This generalizes a recent joint result with Lichtman (where each R; was a domain) and an earlier joint result with Montgomery (where each R, was a domain and ~ was a field). O. Introduction. Let RI and R2 be 6-rings with lover a division ring 6, Rl '* 6, R2 '* 6, and let R = R1UR 2 be the coproduct of Rl and R2 over 6. It is known (and easy to see) that R is a prime ring. Throughout this paper we shall be assuming that at least one of the four dimensions (R i: 6) r' (R i : 6) I' i = 1,2, is greater than 2. Denoting by Z = Z(R) the center of R we know [5, Theorem 1] that Z = Z(R1) n Z(R 2) ~ Z(6). Let R%be the left quotient ring of R relative to the filter '%of all nonzero two-sided ideals of R. Then the set N* = N*(R) of all units of R%such that u-1Ru = R is called the set of normalizing elements for R [9, p. 240] and the automorphisms thus induced on R are just the X-inner automorphisms of Kharachenko [9, p. 3]. An important subset of N = N* u {O} is the extended center C of R. The main goal of this paper is to show that under proper restrictions every X-inner automorphism of R is inner; a precise statement of this result will be given at the end of this introductory section. One restriction of a general nature which we shall make is that the rings R J and R2 be weakly I-finite (i.e., anyone-sided inverse is two-sided). Before describing restrictions of a more specialized nature we must recall a couple of definitions from [5]. A 6-ring S is said to be primary in case S = 6 E9 T, where T is a 6-bimodule such that T2 = O. A 6-ring S is said to be quadratic in case S = 6 E9 6x = 6 E9 x6 for some xES. Situations which we shall eventually want to avoid are of the following three (seemingly artificial) types: (I) Both Rl and R2 are primary. (II) One R i is primary and the other is quadratic. (III) The characteristic of 6 is 2, at least one of the R i is not a domain, and one of the R i is quadratic. Received by the editors February 22,1985. 1980 Mathematics Subject Classification. Primary 16A06; Secondary 16A08. 16A03. 303 @1986 American Mathematical Society 0002-9947/86 $1.00 + $. 25 per page License or copyright restrictions may apply to redistribution; see https://www.ams. org/journal-terms-of-use 304 w. S. MARTINDALE III In a corrected version of [5, Theorems 4, 5] the following result was established: C = Z unless R}, R 2 satisfy (I), (II), or (III). In §I we shall begin by reviewing the definition and basic properties of the key notion of height Irl of an element r E R. These are due primarily to Cohn [3]. Next we shall recall how left and right ~-bases for R} and R2 lead to various types of "monomial" bases for R regarded variously as a left or right ~, R}, or R2 module. We shall sometimes require a more refined notion than that of height; to this end we show that a lexicographic ordering of the "basis" monomials induces a partial well-ordering < on the elements of R. In §2 the units of R are discussed and it seems natural here to make the assumption that R} and R2 are weakly I-finite. The obvious units of R are those generated by the units of R} and R2 (Cohn [4] calls these monomial units) and by units of the form 1 + ajb, a, b E R i, ba = 0, fER (these are the transvections of Bergman [1]). We shall refer to the totality of all such units as standard units. We then develop a result (Theorem 2) which is crucial to our arguments later on. It is too technical to state here with any precision but in a very loose sense it says the following: if Ifgl is not what one would normally expect it to be then there exists a standard unit u such thatfu < f. Among various corollaries is Bergman's result [1, p. 10]: if each Ri is weakly I-finite then every unit of R is a standard unit. The study of X-inner automorphisms is taken up in §3. If a is an X-inner automorphism it is easily seen that there exist nonzero elements f, g, hER such that