Engel varieties of associative rings and the number of Mersenne primes

✍ Scribed by David M. Riley

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 114 KB
• Volume: 261
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00562-8

✦ Synopsis

An associative ring R can be viewed as a semigroup via a • b = a + b + ab, and as a Lie ring via [a, b] = abba. It is known that the Lie ring [R] is nilpotent if and only if the adjoint semigroup (R, •) is nilpotent (in the sense defined by Mal'cev or by Neumann and Taylor). We prove a similar result for associative rings whose Lie rings satisfy an Engel identity. Mersenne primes appear in an unexpected role.