Algebra and Logic Seminar

The next meeting of the seminar will be held on January 26, 2017 (Friday), at 13:00, in Room 578 of the Institute of Mathematics and Informatics.

A talk on

n-Torsion Clean Rings

will be delivered by Peter DANCHEV.

Abstract. 

Recall that an arbitrary ring is called {it clean} by Nicholson (TAMS, 1977) if each its element is the sum of a unit and an idempotent. In addition, if these two elements commute, the clean ring is said to be {it strongly clean}.
In the present talk we define and characterize the following two proper subclasses:

Definition. Let $n$ be an arbitrary natural. We shall say that a ring $R$ is {it $n$-torsion clean} if, for every $rin R$, there exist a unit $u$ with $u^n=1$ and an idempotent $e$ such that $r=u+e$ and $n$ is the smallest possible natural number with the above property. If, in addition, $ue=eu$, $R$ is called {it strongly $n$-torsion clean}.
The main results are as follows:

Theorem 1. If $R$ is a strongly $n$-torsion clean ring, then the following hold:
(1) $R$ is a PI-ring;
(2) $R$ has finite characteristic;
(3) $J(R)$ is a nil-ideal;
(4) $R$ is either abelian or char$(F)$ divides $n$, provided $R$ is an algebra over a field $F$.

Theorem 2. For a ring $R$, the following conditions are equivalent:
(1) There exists $nin mathbb{N}$ such that $R$ is an $n$-torsion clean abelian ring.
(2) char$(R)$ is finite, $J(R)$ is nil of bounded index, idempotents lift uniquely modulo $J(R)$ and $R/J(R)$ is a subdirect product of finite fields $F_i$, where $i$ ranges over some index set $I$, such that $LCM(|F_i|-1mid iin I)$ exists.
(3) $R$ is abelian clean such that $U(R)$ is of finite exponent.

In parallel to the last assertion, the following is true:

Theorem 3. For a ring $R$, the following statements are equivalent:
(1) $R$ is strongly $n$-torsion clean for some $nin mathbb{N}$.
(2) $R$ is strongly clean and $U(R)$ is of finite exponent.

In case that $n$ is odd, we have the following satisfactory structural description:

Theorem 4. Suppose $nin mathbb{N}$ is odd. For a ring $R$, the following points are equivalent:
(1) $R$ is strongly $n$-torsion clean.
(2) There exist integers $k_1,dots ,k_tgeq 1$ such that $n=LCM(2^{k_1}-1, ldots, 2^{k_t}-1)$ and $R$ is a subdirect product of copies of fields $mathbb{F}_{2^{k_i}}$, $1leq ileq t$.
(3) $R$ is clean in which orders of all units are odd, bounded by $n$, and there is a unit of order $n$.

This gives a new recent advantage on the general classification problem for clean rings.

The paper is a joint project with J. Matczuk from Math. Inst. of the Polish Acad. Sci. (Univ. of Warsaw) and will be published by the AMS in a subsequent issue of the Contemp. Math. Series (2017). These results are reported of the conference "Noncommutative Rings and their Applications - V", held in Lens, France, on 12-15 June, 2017.

Everybody is invited.
http://www.math.bas.bg/algebra/seminarAiL/