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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ *$-differential identities of prime rings with involution


Author: Chen-Lian Chuang
Journal: Trans. Amer. Math. Soc. 316 (1989), 251-279
MSC: Primary 16A28; Secondary 16A12, 16A38, 16A72
MathSciNet review: 937242
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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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0937242-2
PII: S 0002-9947(1989)0937242-2
Keywords: Differential identity, generalized (polynomial) identity, left (Martindale) quotient ring, two-sided (Martindale) quotient ring, prime rings with involution
Article copyright: © Copyright 1989 American Mathematical Society