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

Martindale, Wallace S.

doi:10.1090/S0002-9947-1986-0814924-1

The normal closure of the coproduct of rings over a division ring
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by Wallace S. Martindale PDF

Trans. Amer. Math. Soc. 293 (1986), 303-317 Request permission

Abstract:

Let $R = {R_1}\coprod {R_2}$ be the coproduct of $\Delta$-rings ${R_1}$ and ${R_2}$ with 1 over a division ring $\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta$, with at least one of the dimensions ${({R_i}:\Delta )_r}, {({R_i}:\Delta )_l}, i = 1, 2$, greater than 2. If ${R_1}$ and ${R_2}$ are weakly $1$-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 ${R_1}, {R_2}$ satisfy one of the following conditions: (I) each ${R_i}$ is primary (i.e., ${R_i} = \Delta + T, {T^2} = 0$), (II) one ${R_i}$ is primary and the other is $2$-dimensional, (III) char.$\Delta = 2$, one ${R_i}$ is not a domain, and one ${R_i}$ is $2$-dimensional. This generalizes a recent joint result with Lichtman (where each ${R_i}$ was a domain) and an earlier joint result with Montgomery (where each ${R_i}$ was a domain and $\Delta$ was a field).

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