Krull dimension of modules

over involution rings. II

Authors:
K. I. Beidar, E. R. Puczylowski and P. F. Smith

Journal:
Proc. Amer. Math. Soc. **125** (1997), 355-361

MSC (1991):
Primary 16W10, 16P60

DOI:
https://doi.org/10.1090/S0002-9939-97-03724-6

MathSciNet review:
1371115

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a ring with involution and invertible 2, and let be the subring of generated by the symmetric elements in . The following questions of Lanski are answered positively:

- (i)
- Must have Krull dimension when does?
- (ii)
- Is every Artinian -module Artinian as an -module?

**[1]**S.A. Amitsur,*Rings with involution*, Israel J. Math.**6**(1968), 99-106. MR**39:256****[2]**K.I. Beidar, E.R. Puczy{\l}owski and P.F. Smith,*Krull dimension of modules over involution rings*, Proc. Amer. Math. Soc.**121**(1994), 391-397. MR**94h:16064****[3]**K.I. Beidar and V.T. Markov,*A semiprime PI-ring having a faithful module with Krull dimension is a Goldie ring*, Russian Math. Survey**48**(1993), 141-142. MR**94m:16023****[4]**C.L. Chuang and P.H. Lee,*Noetherian rings with involution*, Chinese J. Math.**5**(1977), 15-19. MR**56:12053****[5]**R. Gordon and J.C. Robson,*Krull dimension*, Memoirs Amer. Math. Soc., No. 133, American Mathematical Society, Providence, 1973. MR**50:4664****[6]**I.N. Herstein,*Topics in ring theory*, Univ. Chicago Press, Chicago, 1969. MR**42:6018****[7]**C. Lanski,*On the relationship of a ring and the subring generated by its symmetric elements*, Pacific J. Math.**44**(1973), 581-592. MR**48:331****[8]**C. Lanski,*Chain conditions in rings with involution*, J. London Math. Soc.**9**(1974), 93-102. MR**50:13123****[9]**C. Lanski,*Chain conditions in rings with involution II*, J. London Math. Soc.**18**(1978), 421-428. MR**80a:16024****[10]**C. Lanski,*Gabriel dimension and rings with involution*, Houston Math. J.**4**(1978), 397-415. MR**80a:16025****[11]**P.H. Lee On subrings of rings with involution, Pacific J. Math.**60**(1975), 131-147. MR**53:519****[12]**V.T. Markov,*On PI rings having a faithful module with Krull dimension*(to appear).**[13]**S. Montgomery,*A structure theorem and a positive-definiteness condition in rings with involution*, J. Algebra**43**(1976), 181-192. MR**54:12821****[14]**S. Montgomery,*Fixed rings of finite automorphism groups of associative rings*, Lectures Notes in Math. Vol. 818, Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR**81j:16041****[15]**L.H. Rowen,*On rings with central polynomials*, J. Algebra**31**(1974), 393-426. MR**50:2237**

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Additional Information

**K. I. Beidar**

Affiliation:
Department of Mathematics, Moscow State University, Moscow, Russia

Address at time of publication:
National Cheng–Kung University, Department of Mathematics, Tainan, Taiwan

Email:
t14270@sparc1.cc.ncku.edu.tw

**E. R. Puczylowski**

Affiliation:
Institute of Mathematics, University of Warsaw, Warsaw, Poland

Email:
edmundp@mimuw.edu.pl

**P. F. Smith**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, Scotland

Email:
pfs@maths.gla.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-97-03724-6

Received by editor(s):
August 23, 1995

Additional Notes:
The research of the second author was partially supported by KBN grant 2 P301 035 06.

Communicated by:
Ken Goodearl

Article copyright:
© Copyright 1997
American Mathematical Society