## Commutativity of rings with constraints on commutators. II.(English)Zbl 0955.16028

[For part I see ibid. 5, 123-131 (1985; Zbl 0606.16023).]
The author proves commutativity of an associative ring $$R$$ satisfying one of the following conditions: (1) for each $$x,y\in R$$ there exists a co-monic polynomial $$p(t)\in tZ[t]$$, such that $$[x,y]=[x,y](p(xy)-p(yx))$$; (2) for each $$x,y\in R$$ there exist $$p(t),q(t)\in tZ[t]$$ with $$q(t)$$ of the form $$(t-t^2f(t))$$, such that $$[x,y]=[x,y]p(x)q(y)$$; (3) $$(R,+)$$ is a torsion group and for each $$x,y\in R$$ there exist $$p(t),q(t)\in tZ[t]$$ such that $$[x,y]=[x,y]p(x)q(y)$$ and the coefficients of $$q'(t)$$ have highest common factor 1; (4) for each $$x,y\in R$$ there exists $$n=n(x,y)\geq 1$$ for which $$[y,x]=[y^mx^n,x]$$, where $$m$$ is a fixed positive integer; (5) for each $$x,y\in R$$ either $$[x,y]=0$$ or there exists an integer $$n=n(x,y)>1$$ for which $$xy=x^ny^n$$ and $$yx=y^nx^n$$.

### MSC:

 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras) 16R50 Other kinds of identities (generalized polynomial, rational, involution)

### Keywords:

commutativity theorems; polynomial constraints

Zbl 0606.16023
Full Text:

### References:

  Bell, H. E.: Commutativity of rings with constraints on commutators. Results Math. 8 (1985), 123–131. · Zbl 0606.16023  Herstein, I. N: The structure of a certain class of rings. Amer. J. Math. 75 (1953), 864–871. · Zbl 0051.02501  Herstein, I.N.: Two remarks on the commutativity of rings. Canad. J. Math. 7 (1955), 411–412. · Zbl 0065.02203  Hongan, M. and Tominaga, H:. Some commutativity theorems for semiprime rings. Hokkaido Math. J. 10 (1981), 271–277. · Zbl 0487.16024  Kezlan, T.P.: Another commutativity theorem involving certain polynomial constraints. Math. Japonica 48 (1998), 287–290. · Zbl 0922.16019  Streb, W.: Über einem Satz von Herstein und Nakayama. Rend. Sem. Mat. Univ. Padova 64 (1981), 159–171. · Zbl 0474.16024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.