annihilators of nilpotent elements · 2019. 8. 1. · annihilators of nilpotent elements abraham a....

ANNIHILATORS OF NILPOTENT ELEMENTS

ABRAHAM A. KLEIN

Received 23 February 2005 and in revised form 4 September 2005

Let x be a nilpotent element of an infinite ring R (not necessarily with 1). We prove thatA(x)—the two-sided annihilator of x—has a large intersection with any infinite idealI of R in the sense that card(A(x)∩ I) = cardI . In particular, cardA(x) = cardR; andthis is applied to prove that if N is the set of nilpotent elements of R and R �= N , thencard(R\N)≥ cardN .

For an element x of a ring R, let A�(x), Ar(x), and A(x) denote, respectively, the left,right and two-sided annihilator of x in R. For a set X , we denote cardX by |X|; andsay that a subset Y of X is large in X if |Y | = |X|. We prove that if x is any nilpotentelement and I is any infinite ideal of R, then A(x)∩ I is large in I , and in particular|A�(x)| = |Ar(x)| = |A(x)| = |R|. The last result is applied to obtain a generalization ofa result of Putcha and Yaqub [2] which shows that an infinite nonnil ring has infinitelymany nonnilpotent elements. A short proof of their result is given in [1]. We prove amuch stronger result showing that the set of nonnilpotent elements of a nonnil ring isat least as large as is its set of nilpotent elements. The following lemma is simple butcrucial.

Lemma 1. Let R be an infinite ring, (S,+) an infinite subgroup of (R,+), and x an element ofR. Then either |Sx| = |S| or |A�(x)∩ S| = |S|, and similarly |xS| = |S| or |Ar(x)∩ S| = |S|.Proof. Consider the map y �→ yx from (S,+) onto (Sx,+). The kernel is A�(x)∩ S, so|S| = |Sx||A�(x)∩ S| and the result follows since S is infinite. �

A subset of a ring R is said to be root closed if whenever it contains a power of anelement, it also contains the element itself.

Theorem 2. Let R be an infinite ring and α an infinite cardinal. Then, the following hold.(i) For any left (right) ideal I of R, the set {x ∈ R | |Ar(x)∩ I| = α} (resp., {x ∈ R ||A�(x)∩ I| = α}) is root closed. In particular, if I is infinite, {x ∈ R | |Ar(x)∩ I| =|I|} (resp., {x ∈ R | |A�(x)∩ I| = |I|}) is root closed, so it contains the set N ofnilpotent elements of R.

Copyright © 2005 Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical Sciences 2005:21 (2005) 3517–3519DOI: 10.1155/IJMMS.2005.3517

http://dx.doi.org/10.1155/S0161171205502552

3518 Annihilators of nilpotent elements

(ii) For any ideal I of R, {x ∈ R | |A(x)∩ I| = α} is root closed. In particular, if I isinfinite, {x ∈ R | |A(x)∩ I| = |I|} is root closed, so it contains N .

Proof. (i) Let |Ar(xn)∩ I| = α for some n≥ 2 and consider xn−1(Ar(xn)∩ I). By Lemma 1,either |xn−1(Ar(xn) ∩ I)| = α or |Ar(xn−1) ∩ I| = |Ar(xn−1) ∩ (Ar(xn) ∩ I)| = α. Nowxn−1(Ar(xn) ∩ I) ⊆ Ar(x) ∩ I ⊆ Ar(xn−1) ∩ I ⊆ Ar(xn) ∩ I , so |Ar(xn−1) ∩ I| = α evenwhen |xn−1(Ar(xn)∩ I)| = α. It follows by induction that |Ar(x)∩ I| = α.

(ii) Let |A(xn)∩ I| = α for some n≥2. Since A�(xn)∩ I is a left ideal and |Ar(xn)∩(A�(xn) ∩ I)| = |A(xn) ∩ I| = α, it follows by (i) that |Ar(x) ∩ (A�(xn) ∩ I)| = α =|A�(xn)∩ (Ar(x)∩ I)|; and since Ar(x)∩ I is a right ideal, we get, again by (i), that|A�(x)∩ (Ar(x)∩ I)| = α, namely |A(x)∩ I| = α. �

Applying the previous theorem for I = R, we obtain the following corollary.

Corollary 3. Let x be a nilpotent element of an infinite ring R, then |A�(x)| = |Ar(x)| =|A(x)| = |R|.

The previous corollary will be applied in the proof of the above-mentioned general-ization of a result of Putcha and Yaqub [2]. We also need the following result.

Lemma 4. Let b be a nonnilpotent element of an infinite ring R. If R\N is infinite, then|A�(b)| ≤ |R\N| and |Ar(b)| ≤ |R\N|.Proof. Let x ∈ A�(b)∩N , then xb = 0 and xn = 0 for some n ≥ 1. Let m ≥ n, then (b +x)m = bm + bm−1x + ···+ bxm−1. Since (bm−1x + ···+ bxm−1)2 = 0 and b �∈ N , b2m �= 0and (b + x)m �= 0, so b + x �∈ N . Hence, the map x �→ b + x is 1− 1 from A�(b)∩N intoR\N and therefore |A�(b)∩N| ≤ |R\N|. Since R\N is infinite, we get that |A�(b)| =|A�(b)\N|+ |A�(b)∩N| ≤ |R\N|+ |R\N| = |R\N|. �

In a ring with 1, the map x �→ 1 + x from N into R\N is 1− 1, so |R\N| ≥ |N|. Thenext theorem shows that the same result holds in any nonnil ring. In particular, we getthe result of Putcha and Yaqub [2] stating that R is finite when R\N is finite and notempty.

Theorem 5. Let R be a nonnil ring, then |R\N| ≥ |N|.Proof. We start with R infinite. Suppose |R\N| < |N|, then |N| = |R| and |R\N| < |R|.By the previous lemma, if b ∈ R\N , |A�(b)| ≤ |R\N|, so |A�(b)| < |R| and by Lemma 1,|Rb| = |R|. Now |R| = |Rb| ≤ |Nb|+ |(R\N)b| and |(R\N)b| ≤ |R\N| < |R|, so |Nb| =|R|. Therefore, |{b+ xb|x ∈N}| = |R|, so since |R\N| < |R|, there exists x ∈N such thatb + xb �∈ R\N , namely b + xb ∈ N . Since x ∈ N , 1 + x is formally invertible, so Ar(b +xb)= Ar(b). By Corollary 3, |Ar(b+ xb)| = |R| and by Lemma 4, |Ar(b)| ≤ |R\N| < |R|,a contradiction.

Now let R be finite and let J be its radical. Since J is nilpotent, if a∈ R, a+ J is nilpotentin R/J if and only if a is nilpotent, and if a �∈ N , (a + J)∩N = ∅. Since R/J is a finitesemisimple ring, it has 1, so at least half of its elements are nonnilpotent, hence at leasthalf of the distinct cosets a+ J , a∈ R, do not intersect N , and therefore at least half of theelements of R are not nilpotent, so |R\N| ≥ |N|. �

Abraham A. Klein 3519

References

[1] H. E. Bell and A. A. Klein, On finiteness, commutativity, and periodicity in rings, Math. J.Okayama Univ. 35 (1993), 181–188 (1995).

[2] M. S. Putcha and A. Yaqub, Rings with a finite set of nonnilpotents, Int. J. Math. Math. Sci. 2(1979), no. 1, 121–126.

Abraham A. Klein: Department of Pure Mathematics, School of Mathematical Sciences, TheRaymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

E-mail address: [email protected]

mailto:[email protected]

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

annihilators of nilpotent elements · 2019. 8. 1. · annihilators of nilpotent elements abraham a....

Documents