$*$-differential identities of prime rings with involution

Abstract:

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