Kyungpook Mathematical Journal 2022; 62(2): 213-227
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
The Relation Between Units and Nilpotents
Jeoung Soo Cheon, Tai Keun Kwak*, Yang Lee, Young Joo Seo
Department of Mathematics, Pusan National University, Busan 46241, Korea
e-mail : firstname.lastname@example.org
Department of Data Science, Daejin University, Pocheon 11159, Korea
e-mail : email@example.com
Department of Mathematics, Yanbian University, Yanji 133002, China Institute for Applied Mathematics and Optics, Hanbat National University, Daejeon 34158, Korea
e-mail : firstname.lastname@example.org
Department of Data Science, Daejin University, Pocheon 11159, Korea
e-mail : email@example.com
Received: November 17, 2021; Revised: March 8, 2022; Accepted: March 8, 2022
We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring
Keywords: transitivity of units, right UN-transitive ring, unilpotent-IFP ring, unit, nilpotent, nilradical, NI ring
All rings considered in this article are associative with identity unless otherwise stated. Let
Following , a ring
Example 1.1. (1) Let
2. Transitivity and Unilpotent-IFP Rings
In this section we introduce two kinds of ring properties through which we study the relation between units and nilpotents. The first is related to the transitivity of units on nilpotents under regular group actions and the second is related to the property of inserting units into nilpotent products of elements.
Recall first the following definitions. Let
For a ring
Remark 2.1. (1) Let
(2) Consider the unit-IFP ring
We shall call a ring
Lemma 2.2. A non-reduced ring
Given a ring
Proposition 2.3. Let
and N(R)is a subring of R.
(3) Köthe's conjecture holds for
Further, we claim
(3) This is evident from (2).
The proof for the left UN-transitive case is similar.
Each converse of Proposition 2.3 needs not hold by Remark 2.1(2). The rings below shall provide the motivation for the main argument in this article.
Example 2.4. (1) Let
(2) We follow the construction in [6, Example 1.2(2)] which applies [15, Definition 1.3]. Let
For a field
by the argument in [6, Example 1.2(1)], where
Next we consider a generalized condition of one-sided UN-transitivity by considering "⊆", in place of "=".
Proposition 2.5. (1) For a ring
ab ∈ N(R)for a, b ∈ Rimplies aU(R)b ⊆ N(R);
a ∈ N(R)implies ras ∈ N(R)for all r, s ∈ U(R);
ol(a) ⊆ N(R)for all a ∈ N(R);
or(a) ⊆ N(R)for all a ∈ N(R);
a1⋯ an ∈ N(R)for a1, …, an∈ Rand n ≥ 2, then for all u1, …, un+1 ∈ U(R), u1a1u2a2⋯ unanun+1∈ N(R).
(2) Note that
Based on the facts above, we consider the following as a generalization of not only one-sided UN-transitive rings, but unit-IFP rings and NI rings.
A ring shall be called
Every NI ring
The non-unilpotent-IFP rings below provide useful manner to argue about the unilpotent-IFPness of given rings.
Example 2.6. (1) Consider
(2) Consider the rings constructed in [10, Example 2(2)]. Let
which is not a nilpotent, for all
Based on the structures of rings in Example 2.4, a ring
Theorem 2.7. (1)Every right or left unilpotent-duo ring is unilpotent-IFP.
Ris left unilpotent-duo if and only if it is right unilpotent-duo.
Ris left UN-transitive if and only if it is right UN-transitive. Especially, if Ris non-reduced right UN-transitive, there exists such that
2-(ii) First, if
Next, suppose that
From this, we obtain
(3) Suppose that
The converse of Theorem 2.7(1) does not hold in general by Example 3.8(1) to follow.
Corollary 2.8. Let
(1) The group ring
KGis left unilpotent-duo if and only if it is right unilpotent-duo.
(2) The group ring
KGis left UN-transitive if and only if it is right UN-transitive.
One may ask whether if
Example 2.9. Consider the infinite direct product
Recall that unilpotent-IFP rings need not be NI by Example 1.1(1). We see conditions under which unilpotent-IFP rings may be NI. Note that for a ring
Theorem 2.10. Let
from which we infer that
The proof of the left transitive case can be done by symmetry.
Regarding Theorem 2.10, it is evident that if
Polynomial rings over NI rings need not be NI by Smoktunowicz . But if given a ring
Corollary 2.11. Let
Remark 2.12. Let
Next argue about the actual form of elements in
3. Structure of Unilpotent-IFP Rings
In this section we study the structure of unilpotent-IFP rings as well as the relations between unilpotent-IFP rings and related rings, studying the structures of some kinds of unilpotent-IFP rings which are considered ordinarily in (noncommutative) ring theory.
We first note that the class of unilpotent-IFP rings is not closed under homomorphic images, since every ring is a homomorphic image of a free ring (which is reduced and therefore unilpotent-IFP). But we obtain elementary properties for unilpotent-IFP rings as follows. The direct product of rings
Proposition 3.1. (1) If
(3) Since Λ is finite,
We next study some sorts of unilpotent-IFP rings which are able to provide plentiful information to related studies. For a ring
Proposition 3.2. Let
By the same idea as in the proof of Proposition 3.2, we consider a similar proposition which also provides examples of unilpotent-IFP rings, being concerned with modules.
Proposition 3.3. Let
The following is an application of Proposition 2.5(1). Let
Proposition 3.4. Let
In what follows we consider some conditions under which the set of all nilpotent elements in unilpotent-IFP rings forms a subring, which it is compared with Proposition 2.3.
Proposition 3.5. Let
N(R)is a subring of R, and ab=-bafor all .
N(R)is a commutative subring of R, when Ris of characteristic 2.
(2) It is an immediate consequence of (1).
The following elaborates Proposition 3.5.
Example 3.6. (1) We recall the unit-IFP (and so unilpotent-IFP) ring
(2) We recall the ring
Recall that Köthe's conjecture holds for a given ring
Theorem 3.7. (1) A ring
(2) A ring
Conversely, suppose that
(2) The necessity is obvious. For the converse, let
Regarding Theorem 3.7(1), nilpotents always form a subring in a unilpotent-IFP ring satisfying Köthe's conjecture. Indeed, if
Consider the necessity of Theorem 3.7(2). If a ring
The following elaborates upon the relations among the concepts above.
Example 3.8. (1) There exists a unilpotent-IFP ring that is neither right nor left unilpotent-duo. Let
(2) There exists an IFP ring that is neither right nor left unilpotent-duo. We use the ring in [7, Example 2]. Let
The following diagram shows all implications among the concepts above.
In what follows we consider a condition under which the ring properties mentioned above coincide. Following , a ring
The implications (1)
Recall that a ring
The converse of Proposition 3.10 needs not hold as can be seen by
The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much.
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