*-differential identities of prime rings with involution

Abstract

Main Theorem. Let R R be a prime ring with involution ∗ ^{\ast } . Suppose that ϕ ( x i Δ j , ( x i Δ j ) ∗ ) = 0 \phi (x_i^{{\Delta _j}},{(x_i^{{\Delta _j}})^{\ast }}) = 0 is a ∗ {\ast } -differential identity for R R , where Δ j {\Delta _j} are distinct regular words of derivations in a basis M M with respect to a linear order > > on M M . Then ϕ ( z i j , z i j ∗ ) = 0 \phi ({z_{ij}},z_{ij}^{\ast }) = 0 is a ∗ {\ast } -generalized identity for R R , where z i j {z_{ij}} are distinct indeterminates. Along with the Main Theorem above, we also prove the following: Proposition 1. Suppose that ∗ ^{\ast } is of the second kind and that C C is infinite. Then R R is special. Proposition 2. Suppose that S W ( V ) ⊆ R ⊆ L W ( V ) {S_W}(V) \subseteq R \subseteq {L_W}(V) . Then Q Q , the two-sided quotient ring of R R , is equal to L W ( V ) {L_W}(V) . Proposition 3 (Density theorem). Suppose that D V {}_DV and W D {W_D} are dual spaces with respect to the nondegenerate bilinear form ( , ) (,) . Let v 1 , … , v s , v s ′ , … , v s ′ ∈ V {v_1}, \ldots ,{v_s},\;v_s^\prime , \ldots ,v_s^\prime \in V and u 1 , … , u t , u 1 ′ , … , u t ′ ∈ W {u_1}, \ldots ,{u_t},\;u_1^\prime , \ldots ,u_t^\prime \in W be such that { v 1 , … , v s } \{ {v_1}, \ldots ,{v_s}\} is D D -independent in V V and { u 1 , … , u t } \{ {u_1}, \ldots ,{u_t}\} is D D -independent in W W . Then there exists a ∈ S W ( V ) a \in {S_W}(V) such that v i a = v i ′ ( i = 1 , … , s ) {v_i}a = v_i^\prime \,(i = 1, \ldots ,s) and a ∗ u j = u j ′ ( j = 1 , … , t ) {a^{\ast }}{u_j} = u_j^\prime \,(j = 1, \ldots ,t) if and only if ( v i ′ , u j ) = ( v i , u j ′ ) (v_i’,{u_j}) = ({v_i},u_j’) for i = 1 , … , s i = 1, \ldots ,s and j = 1 , … , t j = 1, \ldots ,t . Proposition 4. Suppose that R R is a prime ring with involution ∗ ^{\ast } and that f f is a ∗ {\ast } -generalized polynomial. If f f vanishes on a nonzero ideal of R R , than f f vanishes on Q Q , the two-sided quotient ring of R R .