ON TORSION-FREE PERIODIC RINGS

We characterize several large classes of periodic rings: periodic rings with identity, finite-rank torsion-free periodic rings, and rank-two torsion-free periodic rings.


Introduction
There is a great deal of literature on periodic rings, respectively, torsion-free rings (especially of rank two).The aim of this paper is to provide a link between these two topics.
All groups considered here are Abelian, with addition as the group operation.By order of an element we always mean the additive order of this element.All rings are associative but not necessarily with identity.The additive group of the ring R will be denoted by R + .ᏹ n (R) denotes the ring of all the n × n matrices with entries in R.
A ring R is called periodic if for each x ∈ R, the set {x, x 2 ,x 3 ,...} is finite, or equivalently, for each x ∈ R there are positive integers m(x), n(x) such that x m(x) = x m(x)+n (x) .However, periodic rings can also be defined (see [20]) by requiring that (i) the multiplicative semigroup of R is periodic, or, (ii) if a ∈ R, then a power of a generates a finite subring.Examples of periodic rings are finite rings, nil rings, and direct sums of matrix rings over finite fields.Z, the ring of all the integers, is not periodic.
Research on periodic rings (the term "periodic" seems to have been first used by Chacron [16]) was mainly done in two directions: (i) finding sufficient conditions on periodic rings which imply commutativity, Bell being the prominent name in this direction (all over the last 40 years; e.g., see [10,11,12]) but also Abu-Khuzam and Yaqub (see [1,2,13,26]), respectively, (ii) finding structure results for some special classes of periodic rings (e.g., see [3,5,12]).However, it should be noticed that the starting point for these investigations was the Jacobson theorem, whose proof contains many ideas which could be used also in more general contexts.
For later convenience we state here some elementary properties for a periodic ring.
(iii) Any infinite-order element is a zero divisor (in the subring generated by itself).
(iv) Every idempotent in R has finite order.(v) For each a ∈ R some power of a is idempotent.
On the other hand, research on the additive groups of rings begun much earlier.Defining ring structures on Abelian groups was first done by Beaumont [6] who considered rings on direct sums of cyclic groups.Nearly at the same time, Szele investigated nil rings [24] and Beaumont and Zuckerman described the rings on subgroups of the rationals.
Satisfactory results were obtained later by Beaumont and Pierce for finite-(and especially 2) rank torsion-free groups, see [7,8].Szele began the program of investigating the additive structures of rings by the study of nilpotent rings (see [25]).However, a complete status of the results (previous to 1973) is given in the Fuchs treatise [19].As of special interest for our paper, we also mention Freedman [18] and Stratton [23] who proved that nonnil torsion-free Abelian groups of rank two possess a unique minimal type, and their typeset has cardinality at most three.Here typeset (R), the typeset of R (or R + ), denotes the set of all types of the elements in R. For the definition of height and type of an element, we refer to [19].For any group G and any type τ, G(τ) = {x ∈ G | t(x) ≥ τ}.For a torsion-free group G, E(G) denotes the endomorphism ring and QE(G) = Q ⊗ Z E(G) the quasiendomorphism ring.
Our main results can be summarized as follows.In Section 2, we determine the structure of the periodic rings with identity.In Section 3, we characterize periodic rings which have a finite-rank torsion-free underlying additive group, obtaining as a by-product a special case confirmation of Köthe's conjecture.In Section 4, we characterize the periodic torsion-free rings of rank two.

Periodic rings with identity
Given any ring R, for any fixed a ∈ R, the left and right multiplications with a are endomorphisms of R + .Therefore, fully invariant subgroups of R + are necessarily ideals in R, no matter how multiplication is defined.
As a special case, the torsion part T(R) is a (two-sided) ideal of R.Moreover, the primary components R p (p prime numbers) of R + are also ideals of R, and every ring with torsion additive group decomposes (as a ring): R = p∈P R p , P denoting the set of all prime numbers.A ring will be called a p-ring (p prime number) if its additive group is an (Abelian) p-group.An Abelian group is bounded if there exists a positive integer n such that nR = {0}.

Definition 2.1. A ring property Λ is called non-Z, if the ring of integers does not have property Λ.
Examples of such properties are Λ ≡ has zero divisors, or, Λ ≡ periodic.
Proposition 2.2.Let R be a ring with identity which satisfies a non-Z property Λ together with its subrings.Then R + is torsion.Moreover, R + is bounded.
Proof.If 1 R denotes the identity, there is a canonical ring homomorphism f : , the ideal generated by the characteristics of R, and im f = 1 Z/ ker f , the subring generated by 1 R .Together with R, 1 Z/ ker f has property Λ and so, ker As for the last claim, if

Corollary 2.3 (see [19]). A structure of ring with (left) identity exists on a torsion group G if and only if G is bounded.
Corollary 2.4.Every periodic ring with identity is torsion (as a group).Moreover, it is bounded, and so, it is a direct sum of cyclic groups.
As a special case, any semisimple periodic ring R is bounded (this will be used in the next section).
Corollary 2.5.Every periodic ring with identity decomposes (as a ring) in a direct sum of p-rings.Each periodic p-ring is (as a group) a direct sum of cyclic p-groups.
Corollary 2.6 (see [21]).A periodic ring with identity such that R + is finitely generated, is finite.
According to Corollary 2.5, the structure of periodic rings with identity reduces to prings which (as groups) are direct sums of cyclic p-groups.A special case of an early result due to Lászl ó Fuchs settles this.
Theorem 2.7 (see [19]).A multiplication µ on a direct sum G = i∈I a i of cyclic pgroups is completely determined by the values µ(a i ,a j ) with a i , a j running over this p-basis of G.Moreover, any choice of µ(a i ,a j ) ∈ G with a i , a j from this p-basis of G-subject to the condition ord(µ(a i ,a j )) ≤ min(ord(a i ),ord(a j ))-extends to a multiplication on G.

The multiplication is associative (commutative) if (and only if) it is associative (commutative) on the p-basis {a
More can be done (this is the last needed step): G being bounded, any element a i0 of maximum order of this p-basis can be taken as identity of a ring, by letting a i0 act as multiplication by 1 on a i0 and by trivial multiplication on the other summands (see [19,Theorem 120.8]).
It should be noted that a function µ : for all a, b, c in G. Further, if G = i∈I H i and H i are fully invariant subgroups of G, multiplications on H i (i ∈ I) extend to multiplications on G (and conversely).
According to [19], an Abelian group is called a nil group if there is no ring structure on G other than the zero-ring.

Torsion-free periodic rings of finite rank
Notice that for an arbitrary ring (denoting by J(R) and Nil(R) the Jacobson and the nilradicals, resp.) the following statements (known as Köthe's conjecture) are equivalent: (i) the upper nilradical contains every nil left ideal; (ii) the sum of two nil left ideals is necessarily nil; (iii) Nil(ᏹ n (R)) = ᏹ n (Nil(R)) for all rings and for all n; (iv) J(R[λ]) = Nil(R)[λ] for all rings R, where λ is an indeterminate commuting with all elements of ring.
From the elementary properties we mentioned in the introduction it follows that any periodic torsion-free ring is nilpotent.Moreover (for an elementary proof see [21]) the following holds.Lemma 3.1.A torsion-free ring is periodic if and only if it is nil.Corollary 3.2.If Köthe's conjecture holds, the matrix ring of a periodic torsion-free ring is also periodic.
Next, recall that if R is a torsion-free ring of finite rank, then QR = Q ⊗ R becomes in a natural way a finite-dimensional Q-algebra (this comes back to Cartan and Eilenberg, see [14] or [19,Section 119]).This is a divisible envelope for R + , and the dimension of QR over Q equals the rank of R + .QR may have an identity even if R does not (actually this happens exactly when there is an element e and an integer n such that ex = nex for all elements x in R).Using the previous lemma it follows that R is a periodic ring if and only if QR is periodic.
The following result shows that in the torsion-free finite rank case, any periodic ring must be nilpotent (the converse obviously also holds).
Theorem 3.3.Let R be a periodic torsion-free ring of rank n.Then R n+1 = 0.
Proof.Since R is periodic, every element of R is nilpotent.Thus, the endomorphisms of the group R + of the form t r : R → R, t r (x) = rx, are nilpotent endomorphisms, hence they belong to N(E(R + )), the nil-radical of the endomorphism ring of R + .But (see [4,Theorem 9.1]) this nil-radical is nilpotent and so there exists a positive integer k > 0 such that t r1 ••• t rk = 0 for any r 1 ,...,r k ∈ R. Therefore R is a nilpotent ring.
Next, if R is a torsion-free ring of finite rank, the finite-dimensional Q-algebra QR = Q ⊗ R is an Artinian Q-algebra.As previously noticed, R is a periodic ring if and only if QR is periodic (indeed, for all s ∈ R, ∃m : r m = 0 implies for all αs ∈ QR (α ∈ Q)∃m : (αr) m = α m r m = 0).
But QR is an n-dimensional Q-algebra, hence every strictly descending chain of Qideals of QR has at most n nonzero terms.Since QR is nilpotent as a periodic ring, we use the chain (QR) ≥ (QR) 2 ≥ ••• ≥ (QR) n+1 , and the fact that if (QR) s = (QR) s+1 , then (QR) s = (QR) k for all k > s, to obtain 0 = (QR) n+1 = QR n+1 .Corollary 3.4.Let R be a torsion-free ring of finite rank.Then R is periodic if and only if R is nilpotent.
In the literature, rings which are finitely generated as rings have been rarely studied.Obviously, if a ring is finitely generated as a group, it is also finitely generated as a ring.
Corollary 3.5.Let R be a periodic ring of finite torsion-free rank.Then it is finitely generated as a ring if and only if it is finitely generated as a group.
Proof.Let n be the rank of R + .If R = r 1 ,...,r m , then Here, a torsion-free group G is strongly indecomposable if whenever 0 = k ∈ Z and kG ⊆ H ⊕ K ⊆ G, then H = 0 or K = 0.
Proof.The (i) case corresponds to (a) in the preceding discussion.In this situation every pure subgroup is a direct summand, hence the kernel ker( f ), for every nilpotent endomorphism f of G, is a direct summand too.Then type(G/ ker( f )) = type(ker( f )) = type(G) and so type(G) is idempotent.The same conclusion can be deduced from [23].
[22,rk 4.3.From the previous proof, notice that if G is a rank n torsion-free group which admits a nontrivial periodic ring multiplication, then G has a nonzero nilpotent endomorphism.Hence, in the n = 2 case, using[22, Theorem 7.1], the quasiendomorphism ring of G must be one of the following matrix rings:(i) ᏹ 2 (Q), or (ii) the ring of all 2 × 2 rational triangular matrices, or (iii) the ring of all 2 × 2 rational triangular matrices with equal diagonal entries.We summarize from [4,Section 3]what we need in the sequel.For a torsion-free group G of rank two, we have the following possible situations:(a) the quasiendomorphism ring of G is isomorphic to ᏹ 2 (Q) if and only if G = H ⊕ K with type(H) = type(K) (i.e., G is homogeneous completely decomposable), (b) the quasiendomorphism ring of G is isomorphic to the ring of all 2 × 2 rational triangular matrices if and only if G = H ⊕ K with type(H) < type(K), (c) the quasiendomorphism ring of G is isomorphic to the ring of all 2 × 2 rational triangular matrices with equal diagonal entries if and only if G is strongly indecomposable, |typeset(G)| = 2, and G has a nilpotent endomorphism.Notice that in all these cases typeset(A) = {τ 1 ,τ 2 } with τ 1 ≤ τ 2 .