A NOTE ON MAL ’ TSEV-NEUMANN PRODUCTS OF RADICAL CLASSES

A radical class R of rings is elementary if it contains precisely those rings whose singly generated subrings are in R. Many examples of elementary radical classes are presented, and all those which are either contained in the Jacobson radical class or disjoint from it are described. There is a discussion of Mal’tsev products of radical classes in general, in which it is shown, among other things, that a product of elementary radical classes need not be a radical class, and if it is, it need not be elementary. Mathematics Subject Classification (2010): 16N80, 08C99


Introduction
The Mal'tsev-Neumann product of classes X and Y is the class X • Y of rings A having an ideal I ∈ X such that A/I ∈ Y.
There are two such products, X • Y and Y • X , and in general they are not equal.This concept was introduced by Mal'tsev [11] for general algebras and about the same time for groups by Neumann [13], so it seems appropriate to associate both names with these products.More usually they have been called Malt'sev products.
For brevity, and as it introduces no ambiguity, we shall call them simply products in the sequel.Mal'tsev [11] proved that for algebras with commuting congruences, if X and Y are varieties, then so is X • Y. (In the general version the role of ideals is played by congruence classes which are subalgebras.)The case of groups was exploited in [13].
The nature of products of radical classes is of some interest, particularly radical classes which have some resemblance to varieties.This seems not to have been directly investigated, though there are a couple of examples in the literature of radical classes which are products of radical subclasses (see the next section).
A radical class R is elementary if it contains a given ring A precisely when it contains all singly generated subrings of A. Such classes are exemplified by the nil radical class N and the class of boolean rings, or, more generally, a semi-simple radical class (SSR-class) [15].The SSR-classes are varieties and elementary radical classes generally are examples of locally equational classes in the sense of Hu [8].
Further information on elementary radical classes can be obtained from [5].
We shall show that the product of two elementary radical classes may be an elementary radical class, a non-elementary radical class or not a radical class at all.Thus such a product need not be a locally equational class, so Mal'tsev's result on products of varieties does not generalize to locally equational classes.We also have a result for radicals in general.If R and U are radical classes then both R • U and U • R are radical classes if and only if they are equal.
All radical classes in this paper are radical classes of associative rings.Our notation and terminology are generally consistent with those of [6] and are identical with those of [5].

Results
While the Mal'tsev product of varieties is always a variety (not just for rings, but for all algebraic structures with permutable congruences; see [11,Teorema 7]) not much is known about Mal'tsev-Neumann products of radical classes.An example of a radical class of abelian groups which is a Mal'tsev-Neumann product of two such is given by Proposition 4.6 of [4] (and via A-radicals this gives an example for rings), while in [12] it is shown that the classes T E • J , where J is the Jacobson radical class, T E is the class of torsion rings with non-zero p-components only for p ∈ E and E is finite, are radical classes.As elementary radical classes are in a sense generalized varieties (see Introduction) it is interesting that the Mal'tsev-Neumann product of two elementary radical classes need not be elementary, even when it is a radical class, as we shall see shortly.We begin, though, with some results about Mal'tsev-Neumann products of radical classes in general.
Proof.As radical classes are closed under extensions R • U is the lower radical class defined by R ∪ U, and for the same reason it contains U • R.

3
The product of two radical classes satisfies "most of" the requirements for a radical class.
(ii) closed under unions of ascending chains of ideals.

Proof. (i) If
. .be a chain of ideals of a ring A with each for all α, while R(I α ) ⊆ I α ∩ J, so each (I α + J)/J, as a homomorphic image of Proof.By Proposition 2.2, both R • U and U • R are homomorphically closed and closed under unions of chains of ideals.Suppose they are equal.Let J be an ideal of a ring A with J and We shall now prove that if V is an SSR-class then N • V is a radical class.
Although the overall plan of the proof mimics that of Freidman, the details are quite different, so we shall present the proof in a fair amount of detail.
Lemma 2.4.Let U be an elementary radical class, A a ring with a local family E of U-subrings.Then A ∈ U.
Proof.If a ∈ A then there is a subring B ∈ E with a ∈ B. But then < a >∈ U and it follows that A ∈ U.
Corollary 2.5.Under the conditions of Lemma 2.4, if A is the union of an ascending chain of U-subrings, then A ∈ U. Lemma 2.6.Let A be a ring with a series Proof.Clearly 0 = N (B 0 ) ⊆ N (B 1 ).If for all β < δ we have then we consider two cases.(2) If δ = γ + 1 for some γ, then for α < δ we have either α = γ or α < γ.Lemma 2.8.Let V be an SSR-class, A a ring with N (A) = 0.If A has an ideal We shall call a series with the properties of the one in Lemma 2.6 a (V, N )-series for a SSR-class V if all its factors are in V ∪ N .Lemma 2.9.If where C α = (B α + N (A))/N (A) for each α.We have C α C α+1 for every α and when δ is a limit, Also for each α we have Thus the C α form a (V, N )-series for A. By Corollary 2.7, N (C α ) = 0 for each α.

Proof. (i) If
The converse is given by Lemma 2.9 .) is a four-element field.Thus all singly-generated subrings of M 2 (K(2)) are in N • V AR(K(4)).But as M 2 (K(2)) is simple and neither nil nor commutative, it is not in N •V AR(K(4)).
On the other hand, the radical class N • B is elementary.This follows from our next theorem, which makes use of a ring theoretic property we now introduce.
A ring A is (strongly) nil-clean [1] if for each a ∈ A there exist an idempotent e and a nilpotent element z such that a = e + z (and ez = ze).
It has recently been shown [9] that the strongly nil-clean rings are precisely the rings in N • B. The (strongly) nil-clean property is a generalization of the (strongly) clean property [1].All of these properties have been mostly studied for rings with identity; the paper of Nicholson and Zhou [14] extended cleanness to rings generally, and in subsequent papers some attention has been given to the wider setting.
(the join of the two varieties).Also V ∨W ⊆ V • W as the latter is a variety containing V and W.But if R ∈ V • W then V(R) ∈ T {p:p|m} and R/V(R) ∈ T {p:p|n} .But m and n are relatively prime, so for additive groups we have Ext((R/V(R)) + , V(R) + ) = 0 (see, e.g.[3], p.267, (C), (D)), whence (where R + p is the p-component of R + ).But the two direct sums on the RHS are (the additive groups of) ideals of R, so R = T {p:p|m} (R) ⊕ T {p:p|n} (R) = V(R) ⊕ W(R).
We now have Note that V•W is again a SSR-class.Whether there are"non-splitting" examples of "commuting pairs" of radical classes we do not know.
under extensions and is therefore a radical class.The converse follows from Lemma 2.1.We shall call a set E of subrings of a ring A a local family if E is up-directed by inclusion and A = {B : B ∈ E}.The set of all finitely generated subrings is always a local family, though the set of one-generator subrings generally is not.Freidman [2] considered a hereditary supernilpotent radical class R with the following extra property.Local Radical Condition.If a ring A has a local family of R-subrings, then A is itself in R. Radical classes satisfying this condition include the locally nilpotent, Jacobson and Brown-McCoy radical classes and N .Freidman showed that if R satisfies the Local Radical Condition, then R • C is also a radical class satisfying this condition, where C is the class of commutative rings.

(
ii) By Proposition 2.11, N • V is closed under extensions.The other requirements for a radical class are given by Proposition 2.2 or by Proposition 2.11 and Corollary 2.13.If B is a subring of a ring A∈ N • V, then B ∩ N (A) ∈ N and B/B ∩ N (A) ∼ = (B + N (A))/N (A) ⊆ A/N (A) ∈ V so B ∈ N • V.The condition involving ( * ) is just Lemma 2.12.Condition ( * ) is weaker than the elementary property: for instance the locally nilpotent radical class L satisfies ( * ), but N is the smallest elementary radical class containing L. Moreover, N • V need not be elementary.Example 2.15.N • V AR(K(4)) is not an elementary radical class.We'll show this by examining M 2 (K(2)).It is a routine matter to show that all one-generator subrings of M 2 (K(2)) except those generated by 1