Goldie conditions in finite normalizing extensions

Lanski, Charles

doi:10.1090/S0002-9939-1980-0572292-7

Goldie conditions in finite normalizing extensions
HTML articles powered by AMS MathViewer

by Charles Lanski

Proc. Amer. Math. Soc. 79 (1980), 515-519

DOI: https://doi.org/10.1090/S0002-9939-1980-0572292-7

PDF | Request permission

Abstract:

Let S be a finite normalizing extension of a ring R. If M is an S module, then M has finite uniform dimension if and only if it has finite uniform dimension when considered as an R module. Consequently, when S is a right Goldie ring, R is also a right Goldie ring. Conversely, if R is a semiprime right Goldie ring and S is a prime ring, then S is a Goldie ring. Finally, when both S and R are semiprime right Goldie rings, the quotient ring of R embeds in the quotient ring of S.

References

Edward Formanek and Arun Vinayak Jategaonkar, Subrings of Noetherian rings, Proc. Amer. Math. Soc. 46 (1974), 181–186. MR 414625, DOI 10.1090/S0002-9939-1974-0414625-5
Joe W. Fisher, Charles Lanski, and Jae Keol Park, Gabriel dimension of finite normalizing extensions, Comm. Algebra 8 (1980), no. 16, 1493–1503. MR 585202, DOI 10.1080/00927878008822529
B. Lemonnier, Dimension de Krull et codéviation. Application au théorème d’Eakin, Comm. Algebra 6 (1978), no. 16, 1647–1665 (French). MR 508242, DOI 10.1080/00927877808822313
J. C. Robson and Lance W. Small, Liberal extensions, Proc. London Math. Soc. (3) 42 (1981), no. 1, 87–103. MR 602124, DOI 10.1112/plms/s3-42.1.87