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Medial rings and an associated radical. (English) Zbl 0732.16011
The authors study certain classes of associative polynomial identity rings R, namely:
medial rings: $$R[R,R]R=0$$, where $$[R,R]=\{xy-yx$$, for all x,y in $$R\}$$ ;
left permutable rings: $$[R,R]R=0;$$
right permutable rings: $$R[R,R]=0$$; and
permutable rings: $$R[R,R]=0=[R,R]R.$$
These classes generalize the class of commutative rings. Examples establish that all 4 classes are distinct. There are many interesting results but the main emphasis is on medial rings, e.g. Theorem 2.5 If R is medial then there exists a commutative right ideal B of R which is maximal among reduced (i.e. no nonzero nilpotent elements) right ideals of R such that $$B\oplus N$$ (where N is the set of all nilpotent elements of R) is an ideal of R which is essential as a right ideal of R. Proposition 3.7 If R is the ring of $$n\times n$$ matrices over a ring K, with $$n\geq 2$$, then R is medial if and only if $$K^ 4=0$$. Furthermore R is left (right) permutable if and only if $$K^ 3=0.$$
For an arbitrary ring R, the medial quasi-radical of R, $$M(R)=the$$ ideal of R generated by R[R,R]R. Then R/M(R) is medial. A ring A is a radical ring if $$M(A)=A$$ and the radical $${\mathcal M}(R)$$ of any ring R is defined as the sum of all the radical ideals of R. This radical has the same class of semisimple rings as the upper radical determined by the class of all medial rings.
There are some intricate structure theorems and examples including the fact that R is a right essential extension of $${\mathcal M}(R)+{\mathfrak L}(R)$$, where $${\mathfrak L}(R)=the$$ left annihilator in R of [R,R]R and is a medial ideal of R. Furthermore, $${\mathfrak L}(R)$$ is a right essential extension of an ideal T of R and T is right permutable. And T itself is a left essential extension of a permutable ideal of R. See diagram 4.4. in the paper. The authors also prove Proposition 5.3 If R is medial then its nil radical equals its prime radical. There are interesting results on medial rings with D.C.C. and on medial rings with A.C.C. (Theorems 6.3, 6.5 and 6.7). The paper concludes with a fine classification of medial subdirectly irreducible rings.

##### MSC:
 16N80 General radicals and associative rings 16D25 Ideals in associative algebras 16R40 Identities other than those of matrices over commutative rings 16U80 Generalizations of commutativity (associative rings and algebras) 16D80 Other classes of modules and ideals in associative algebras
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