Nil Polynomials of Prime Rings

✍ Scribed by Chi-Tsuen Yeh; Chen-Lian Chuang

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 153 KB
• Volume: 186
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0394

✦ Synopsis

Ž . I RESULT

Let R be an associative ring. An element r g R is said to be nilpotent if r n s 0 for some integer n G 1. A subset S of R is called nil if all r g S are nilpotent. It is easy to see that R has no nil right ideals if and only if R has no nil left ideals. Nil right ideals or nil left ideals together are generally called nil one-sided ideals.

Assume that R is a prime ring with the extended centroid C. The ring

mial in the noncommuting variables x , . . . , x and with the coefficients in

the extended centroid C. The polynomial f is called an identity of R, if Ž . f r , . . . , r s 0 for all r , . . . , r g R. The polynomial f is said to be nil

r s 0 for some integer n s n r , . . . , r G 1 depending on 1 d 1 d r , . . . , r . Our aim is to investigate nil polynomials on a prime ring R 1 d

without nonzero nil one-sided ideals: Polynomial identities are nil on R in a trivial way. One may thus wonder whether the converse holds. By the w x result of 2 , this is false when R is a finite matrix ring over a finite field. 781