Conditions that Guarantee that all Nilpotents Commute with Every Element of an Alternative Ring

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Mathematics (HMC)

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Below I examine the meaning of condition (*) in any (not necessarily associative) ring and show that under rather mild restrictions (e.g.,
2a 2 = 0 and a ε N imply a2 = 0) (*) is equivalent to (**) either every element of N commutes with every element of R, or if this latter does not hold, a2 = 0 for every a ε N, and every element of N anti-commutes with every element of R. In fact, (**) holds if and only if R R satisfies (*). A left identity element in a ring R that satisfies (*) is unique, and if R is alternative, then (*) implies that N is always an ideal of R. A variety of additional assumptions on R guarantee that (*) implies that every element of N commutes with every element of R, but there is an associative ring with identity that satisfies (*) in which some element of N fails either to commute or anti-commute with some element of R.

Then, I apply these results to obtain a commutativity theorem which avoids chain conditions on R/N (and the redundant assumption that N is an ideal of R). Also, some results on anti-commutative rings that satisfy (*) are obtained.

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© 1977 Birkhauser Verlag

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