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

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Abstract
Let R = R 1 ∐ R 2 R = {R_1}\coprod {R_2} be the coproduct of Δ \Delta -rings R 1 {R_1} and R 2 {R_2} with 1 over a division ring Δ , R 1 ≠ Δ , R 2 ≠ Δ \Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta , with at least one of the dimensions ( R i : Δ ) r , ( R i : Δ ) l , i = 1 , 2 {({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2 , greater than 2. If R 1 {R_1} and R 2 {R_2} are weakly 1 1 -finite (i.e., one-sided inverses are two-sided) then it is shown that every X X -inner automorphism of R R (in the sense of Kharchenko) is inner, unless R 1 , R 2 {R_1},\,{R_2} satisfy one of the following conditions: (I) each R i {R_i} is primary (i.e., R i = Δ + T , T 2 = 0 {R_i} = \Delta + T,\,{T^2} = 0 ), (II) one R i {R_i} is primary and the other is 2 2 -dimensional, (III) char. Δ = 2 \Delta = 2 , one R i {R_i} is not a domain, and one R i {R_i} is 2 2 -dimensional. This generalizes a recent joint result with Lichtman (where each R i {R_i} was a domain) and an earlier joint result with Montgomery (where each R i {R_i} was a domain and Δ \Delta was a field).

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