-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

DOI:
https://doi.org/10.1090/S0002-9947-1989-0937242-2

MathSciNet review:
937242

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: **Main Theorem.** *Let* *be a prime ring with involution* . *Suppose that* *is a* -*differential identity for* , *where* *are distinct regular words of derivations in a basis* *with respect to a linear order* *on* . *Then* *is a* -*generalized identity for* , *where* *are distinct indeterminates*.

Along with the Main Theorem above, we also prove the following:

**Proposition 1.** *Suppose that* *is of the second kind and that* *is infinite. Then* *is special*.

**Proposition 2.** *Suppose that* . *Then* , *the two-sided quotient ring of* , *is equal to* .

**Proposition 3** (Density theorem). *Suppose that* *and* *are dual spaces with respect to the nondegenerate bilinear form* . *Let* *and* *be such that* *is* -*independent in* *and* *is* -*independent in* . *Then there exists* *such that* *and* *if and only if* *for* *and* .

**Proposition 4.** *Suppose that* *is a prime ring with involution* *and that* *is a* -*generalized polynomial. If* *vanishes on a nonzero ideal of* , *than* *vanishes on* , *the two-sided quotient ring of* .

**[1]**I. N. Herstein,*Rings with involution*, The University of Chicago Press, Chicago, Ill.-London, 1976. Chicago Lectures in Mathematics. MR**0442017****[2]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[3]**V. K. Harčenko,*Differential identities of prime rings*, Algebra i Logika**17**(1978), no. 2, 220–238, 242–243 (Russian). MR**541758****[4]**V. K. Harčenko,*Differential identities of semiprime rings*, Algebra i Logika**18**(1979), no. 1, 86–119, 123 (Russian). MR**566776****[5]**Charles Lanski,*Differential identities in prime rings with involution*, Trans. Amer. Math. Soc.**291**(1985), no. 2, 765–787. MR**800262**, https://doi.org/10.1090/S0002-9947-1985-0800262-9**[6]**J. Lambek,*Lectures in rings and modules*, Chelsea, 1976.**[7]**Wallace S. Martindale III,*Prime rings with involution and generalized polynomial identities*, J. Algebra**22**(1972), 502–516. MR**0306245**, https://doi.org/10.1016/0021-8693(72)90164-0**[8]**Louis Halle Rowen,*Generalized polynomial identities*, J. Algebra**34**(1975), 458–480. MR**0371945**, https://doi.org/10.1016/0021-8693(75)90170-2**[9]**Louis Halle Rowen,*Polynomial identities in ring theory*, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**576061**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
16A28,
16A12,
16A38,
16A72

Retrieve articles in all journals with MSC: 16A28, 16A12, 16A38, 16A72

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0937242-2

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