A generalization of WUU rings
We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a common generalization of the so-called UU rings examined by Danchev...
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|---|---|
| author | Danchev, P. V. Данчев, П. В. |
| author_facet | Danchev, P. V. Данчев, П. В. |
| author_sort | Danchev, P. V. |
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| datestamp_date | 2019-12-05T09:26:39Z |
| description | We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean
rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a
common generalization of the so-called UU rings examined by Danchev – Lam in (Publ. Math. Debrecen, 2016) and of
the so-called WUU rings studied by the author in (Tsukuba J. Math., 2016). |
| first_indexed | 2026-03-24T02:12:44Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
P. V. Danchev (Plovdiv State Univ., Bulgaria)
A GENERALIZATION OF WUU RINGS
УЗАГАЛЬНЕННЯ WUU КIЛЕЦЬ
We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean
rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a
common generalization of the so-called UU rings examined by Danchev – Lam in (Publ. Math. Debrecen, 2016) and of
the so-called WUU rings studied by the author in (Tsukuba J. Math., 2016).
Визначено клас UNI кiлець та наведено результати їх детальних дослiджень у зв’язку з чистими кiльцями, тобто
основний результат статтi описує комутативнi чистi UNI кiльця з точнiстю до iзоморфiзму. Це нове поняття є
типовим узагальненням так званих UU кiлець, що вивчалися Данчевим та Ламом в (Publ. Math. Debrecen, 2016),
та так званих WUU кiлець, що вивчалися Данчевим в (Tsukuba J. Math., 2016).
1. Introduction and background. Throughout the current article all rings into consideration shall
be assumed to be associative, containing identity element 1 which differs from the zero element 0.
Standardly, U(R) denotes the set of all units of R, \mathrm{I}\mathrm{n}\mathrm{v}(R) the set of all involutions of R, \mathrm{I}\mathrm{d}(R) the
set of all idempotents of R and \mathrm{N}\mathrm{i}\mathrm{l}(R) the set of all nilpotents of R. Moreover, J(R) stands for the
Jacobson radical of R and Z(R) for the center of R. All other notions and notations, not explicitly
introduced herein, may be found in [8]. For instance, accounting to [9], a ring R is called clean if
R = U(R) + \mathrm{I}\mathrm{d}(R), and nil-clean if R = \mathrm{N}\mathrm{i}\mathrm{l}(R) + \mathrm{I}\mathrm{d}(R), accounting to [6]. Imitating [1], a ring
R is called weakly nil-clean, provided that R = \mathrm{N}\mathrm{i}\mathrm{l}(R) \pm \mathrm{I}\mathrm{d}(R). Besides, a ring R is said to be
exchange if, for every a \in R, there exists an idempotent e \in aR such that 1 - e \in (1 - a)R. Clean
rings are always exchange, while the converse is false – notice that for Abelian rings (that are rings
whose idempotents are central) these two concepts are tantamount.
In [5] a ring R is called UU if all units are unipotents, that is, U(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R) + 1. This is
obviously equivalent to the equality U(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R) - 1. On the other side, in [3] were defined the
so-named WUU rings that are rings R whose U(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R)\pm 1 and which are a common extension
of UU rings.
The objective of this article is to generalize these WUU rings by using involutions that are units v
whose square is 1, i.e., v2 = 1; e.g., \pm 1 is so. We call them UNI rings. Resultantly, we shall
establish a complete description of nil-clean UNI rings (Proposition 2.4) and commutative clean UNI
rings (Theorem 2.1). These two assertions are stated in the next section. We close the work with
stating a series of unanswered problems.
2. UNI rings. The next notion is our starting point of view.
Definition 2.1. A ring R is called UNI if , for each u \in U(R), there are n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and
i \in \mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R) such that u = n + i. This is obviously tantamount to U(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R) +
+ \mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R) =
\bigl(
\mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R)
\bigr) \bigl(
1 + \mathrm{N}\mathrm{i}\mathrm{l}(R)
\bigr)
.
The following are non trivial examples of UNI rings and of non UNI rings, thus illustrating that
there is a plenty of them:
c\bigcirc P. V. DANCHEV, 2017
1422 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
A GENERALIZATION OF WUU RINGS 1423
Example 2.1. (a) Any WUU ring, that is a ring R with U(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R)\pm 1, is UNI. In particular,
for any s \in \BbbN , the upper triangular s\times s matrix ring \BbbT s(\BbbZ 2) is UNI.
(b) For any ordinal \mu the direct product B\times
\prod
\mu \BbbZ 3, where for instance B \subseteq
\prod
\lambda \BbbZ 2 is a Boolean
ring for some ordinal \lambda , is UNI. In addition, the direct product \BbbZ 3 \times \BbbZ 3 is UNI but not WUU.
(c) Paralleling to [3], for any s \in \BbbN \setminus \{ 1\} , the upper triangular s \times s matrix ring \BbbT s(\BbbZ 3)
is not UNI and hence not WUU. However, contrasting with that, the equality U(\BbbT s(\BbbZ 3)) =
= \mathrm{N}\mathrm{i}\mathrm{l}(\BbbT s(\BbbZ 3)) + \mathrm{I}\mathrm{n}\mathrm{v}(\BbbT s(\BbbZ 3)) (eventually) holds, but the involutions are not necessarily central.
In fact, if for instance s = 2, the matrices
\biggl(
1 1
0 - 1
\biggr)
and
\biggl(
1 0
0 - 1
\biggr)
are both involutions
satisfying
\biggl(
1 1
0 - 1
\biggr)
=
\biggl(
0 1
0 0
\biggr)
+
\biggl(
1 0
0 - 1
\biggr)
, where the first term
\biggl(
0 1
0 0
\biggr)
is nilpotent of
index 2, and furthermore simple matrix computations show that these two involutions do not lie in
Z(\BbbT 2(\BbbZ 3)), because
\biggl(
1 0
0 - 1
\biggr) \biggl(
0 1
0 0
\biggr)
=
\biggl(
0 1
0 0
\biggr)
\not =
\biggl(
0 - 1
0 0
\biggr)
=
\biggl(
0 1
0 0
\biggr) \biggl(
1 0
0 - 1
\biggr)
as
well as
\biggl(
1 1
0 - 1
\biggr) \biggl(
0 1
0 0
\biggr)
=
\biggl(
0 1
0 0
\biggr)
\not =
\biggl(
0 - 1
0 0
\biggr)
=
\biggl(
0 1
0 0
\biggr) \biggl(
1 1
0 - 1
\biggr)
.
(d) For any s \in \BbbN \setminus \{ 1\} , the full s\times s matrix rings \BbbM s(\BbbZ 2) and \BbbM s(\BbbZ 3) are not UNI.
In some cases the reverse of item (a) holds. Specifically, the following is true.
Proposition 2.1. If R is a ring of characteristic 2, or R is a ring with all nilpotents central for
which 2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), then the notions UNI , WUU and UU do coincide.
Proof. Firstly, suppose 2 = 0 in R. Write u = n+i for some n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and i \in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R),
and lifting by 2, we deduce that u2 = q+1 for some q \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Consequently, u2 - 1 = (u - 1)2 \in
\in \mathrm{N}\mathrm{i}\mathrm{l}(R) implies that u \in 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R), as needed.
Secondly, let R be a ring with 2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) \subseteq Z(R). Since for every i \in \mathrm{I}\mathrm{n}\mathrm{v}(R) we have
(1 - i)2 = 2(1 - i), we obtain by induction that (1 - i)m+1 = 2m(1 - i) for all m \in \BbbN . Hence
1 - i \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and thus \mathrm{I}\mathrm{n}\mathrm{v}(R) \subseteq 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R) which ensures that U(R) \subseteq \mathrm{N}\mathrm{i}\mathrm{l}(R) + \mathrm{I}\mathrm{n}\mathrm{v}(R) \subseteq
\subseteq 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R), as required.
Remark 2.1. It is worth noticing that rings R whose \mathrm{N}\mathrm{i}\mathrm{l}(R) \subseteq Z(R) are studied in details
in [11].
Lemma 2.1. Let R be a UNI ring. Then the following are true:
(1) 2 \in U(R) \Leftarrow \Rightarrow 3 \in \mathrm{N}\mathrm{i}\mathrm{l}(R);
(2) 3 \in U(R) \Leftarrow \Rightarrow 2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R).
Proof. For any a \in U(R) one writes that a = n+ i, where n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and i \in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R),
and squaring this equality we see that a2 = 1+ (n2 + 2ni) = 1 + q, where q = n2 + 2ni \in \mathrm{N}\mathrm{i}\mathrm{l}(R),
because n2 and 2ni are commuting nilpotents.
(1) \Rightarrow . Since by what we have shown above 4 = 22 = 1 + q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), it follows at once that
3 \in \mathrm{N}\mathrm{i}\mathrm{l}(R).
\Leftarrow . Since 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R) \subseteq U(R), it follows that 22 = 4 = 1 + 3 \in U(R), i.e., 2 \in U(R).
(2) \Rightarrow . Since by what we have proven above 9 = 32 = 1 + q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), it follows at once that
8 = 23 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), that is, 2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R).
\Leftarrow . Since 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R) \subseteq U(R), as desired it follows that 1 + 2 = 3 \in U(R).
Remark 2.2. If R is a commutative UNI ring with 1 \in U(R) + U(R), then 3 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), i.e.,
2 \in U(R).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1424 P. V. DANCHEV
In fact, writing 1 = n1+ i1+n2+ i2 for some n1, n2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and i1, i2 \in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R), and
squaring this, one deduces that 2i1i2 = - 1 - q for some q \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Again lifting the last equality
by 2, we infer that 4 = 1 + d for some d \in \mathrm{N}\mathrm{i}\mathrm{l}(R) which gives 3 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as stated.
Proposition 2.2. If R is a UNI clean ring, then 6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R).
Proof. Write 3 = u + e = n + i + e for some u \in U(R), e \in \mathrm{I}\mathrm{d}(R), n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and
i \in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R). Hence 3 - i = n+ e and thus ne = en. Therefore, 3 - i is a strongly nil-clean
element, so that (3 - i)2 - (3 - i) \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and 7 - 5i \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Further, 5i = 7 - q for some
q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and lifting that by 2 we routinely conclude that 24 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) whence 63 = 24.9 \in \mathrm{N}\mathrm{i}\mathrm{l}(R)
guarantees that 6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as asserted.
Proposition 2.3. The next three issues hold:
(i) Let R be a UNI ring. Then J(R) is nil and R/J(R) is a UNI ring.
(ii) Let R be a commutative ring of prime characteristic. Then R is a UNI ring if and only if
J(R) is nil and R/J(R) is a UNI ring.
(iii) Let R be an Abelian ring with 3 \in J(R). Then R is a UNI ring if and only if J(R) is nil
and R/J(R) is a UNI ring.
Proof. (i) Given j \in J(R) and since 1 + J(R) \subseteq U(R), one writes that 1 + j = n + i for
n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and i \in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R). Lifting this equality by 2, we have 2j+ j2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Similarly,
- 2j + j2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) by considering 1 - j \in U(R). Since 2j + j2 and - 2j + j2 commute, their sum
2j2 again lies in \mathrm{N}\mathrm{i}\mathrm{l}(R). Since j2 \in J(R) and 1 + j2 \in U(R), by continuing with the same trick
we derive that 2j2 + j4 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) whence j4 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Finally j \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as expected.
Dealing with the second half-part, it follows easily taking into account the well-known critical
fact that the map U(R) \rightarrow U(R/J(R)), induced by the natural map R \rightarrow R/J(R), is always
surjective due to one of the arguments that either 1+ J(R) \leq U(R) or that J(R) is nil. In addition,
U(R/J(R)) \sim = U(R)/(1 + J(R)).
(ii) Suppose now that R is commutative of prime characteristic, say p. One way follows from
point (i) above, so we consider the reciprocal. Since the quotient R/J(R) is commutative UNI
semiprimitive, and hence reduced, each its unit must be an involution. Given u \in U(R), it plainly
follows that u + J(R) is a unit in R/J(R). Therefore, (u + J(R))2 = u2 + J(R) = 1 + J(R)
giving that u2 - 1 \in J(R). Firstly, if char(R) = 2, we obtain that (u - 1)2 \in J(R) \subseteq \mathrm{N}\mathrm{i}\mathrm{l}(R). Thus
u - 1 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), so that u \in 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R). That is why, R is UU whence UNI.
Assume now that char(R) = p is odd. As above, write u2 = 1 + z for some z \in J(R). Since
we have that J(R) is nil, it must be that zk = 0 and so zp
k
= 0 for some natural number k. So,
(u2)p
k
= (1+z)p
k
= 1+zp
k
= 1, i.e., (up
k
)2 = 1. We furthermore write that u = up
k
u - (pk+1)(1+z).
But pk + 1 is always even, so that u - (pk+1)(1 + z) \in 1 + J(R) \subseteq 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R), because u - 2 =
= (1 + z) - 1 \in 1 + J(R). Consequently, U(R) = \mathrm{I}\mathrm{n}\mathrm{v}(R)(1 + \mathrm{N}\mathrm{i}\mathrm{l}(R)), as required.
(iii) One way follows again from point (i) above, so we treat the converse. Now, supposing that
u \in U(R), it simple follows that u+J(R) is a unit of R/J(R), which factor-ring is Abelian as well,
because J(R) is nil. Since U(R/J(R)) \subseteq \mathrm{I}\mathrm{n}\mathrm{v}(R/J(R)) + \mathrm{N}\mathrm{i}\mathrm{l}(R/J(R)), we write u + J(R) =
= (w + J(R)) + (q + J(R)), where (w + J(R))2 = 1 + J(R) and (q + J(R))m = J(R) for some
m \in \BbbN . Since J(R) is nil, it readily follows that q \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Moreover, w2 + J(R) = 1 + J(R)
leads us to ( - w - 1 + J(R))2 = ( - w - 1)2 + J(R) = w2 + 2w + 1 + J(R) = 2w + 2 + J(R) =
= - w - 1+3w+3+J(R) = - w - 1+J(R), that is, ( - w - 1)2 - ( - w - 1) \in J(R). Since J(R)
is nil, there exists e \in \mathrm{I}\mathrm{d}(R) such that e+J(R) = - w - 1+J(R), i.e., w+J(R) = - 1 - e+J(R).
Therefore, one writes that u+ J(R) = - 1 - e+ q + J(R) and thus u = (2e - 1) + ( - 3e+ z) + q
for some z \in J(R). But (2e - 1)2 = 1 and z - 3e \in J(R), so that q + ( - 3e + z) \in \mathrm{N}\mathrm{i}\mathrm{l}(R)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
A GENERALIZATION OF WUU RINGS 1425
because it is not too hard to verify that J(R) +\mathrm{N}\mathrm{i}\mathrm{l}(R) = \mathrm{N}\mathrm{i}\mathrm{l}(R), provided J(R) \subseteq \mathrm{N}\mathrm{i}\mathrm{l}(R). Finally,
u \in (\mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R)) + \mathrm{N}\mathrm{i}\mathrm{l}(R), because the idempotent e commutes with all elements of R.
We thus come to the following:
Remark 2.3. In contrast to UU rings and same as WUU rings (see, e.g., [5] and [3], respectively)
the reverse claim in Proposition 2.3 may fail in general: in fact, as in [3] and [5], there is a nil ideal
I = \{ (aij) \in \BbbT 2(\BbbZ 3) : \forall aii = 0\} of \BbbT 2(\BbbZ 3) such that \BbbT 2(\BbbZ 3)/I \sim = \BbbZ 3 \times \BbbZ 3. However, as pointed
out in Example 2.1 (b) above, the direct product \BbbZ 3 \times \BbbZ 3 is UNI but not WUU (compare with
Remark 2.5 below, too) and, as indicated in Example 2.1 (c), straightforward calculations show that
the ring \BbbT 2(\BbbZ 3) is not UNI. The reason for this is that although I equals to the radical of Jacobson
for \BbbT 2(\BbbZ 3), in accordance with Example 2.1 (c), \BbbT 2(\BbbZ 3) need not be Abelian.
The following technicality is our critical tool (for a weaker variant of this claim the interested
reader may see also [10]).
Lemma 2.2. Suppose u is a unit and e is an idempotent of a ring R such that u2e = eu2 and
u = e+ q or u = - e+ q, where q is a nilpotent. Then e = 1.
In particular, any involution of a weakly nil-clean ring is unipotent.
Proof. Letting u = e+q, for some e \in \mathrm{I}\mathrm{d}(R) and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) with qt = 0, t \in \BbbN say, we obtain
that u2 = e+ eq+ qe+ q2 and hence u2e = e+ eqe+ qe+ q2e. Similarly, eu2 = e+ eq+ eqe+ eq2
and thus u2e = eu2 insures that (q + q2)e = e(q + q2). Thus e commutes with the nilpotent
(q + q2)n = [q(1 + q)]n = qn(1 + q)n for all n \in \BbbN , and therefore the same is valid for u.
Furthermore, u - (q + q2) = e - q2 with u - (q + q2) = u(2) = e - q2 being a unit, one sees that
u(2) - (2q3+ q4) = e - (q2+2q3+ q4) = e - (q+ q2)2. Putting u(3) = u(2)+(q+ q2)2, we observe
that u(3) is a unit since u(2) commutes with (q+ q2)2 and that u(3) = e+ q3(2 + q). Since u(3) will
commute with 2(q+q2)3, it follows that u(4) = u(3) - 2(q+q2)3 = e+f(q4, q5, q6) = e+q4 \cdot h(q, q2),
where f, h are functions of powers of q. Repeating the same procedure t-times, we will find a unit
u(t) such that u(t) = e + qt \cdot a = e for some element a \in R depending on q; if t = 2 we just put
a = - 1 = - q0. This yields that e = 1, which exhausts this case.
Analogously, (q2 - q)e = e(q2 - q) and (q2 - q)u = u(q2 - q). Hence u - q2 = (q - q2) - e.
However, the same trick as above successfully works to conclude the claim after all.
The second part is now immediate.
We now have at our disposal the needed information for classifying (weakly) nil-clean UNI rings
as follows:
Proposition 2.4. (1) A ring R is UNI nil-clean if and only if J(R) is nil and R/J(R) is a
Boolean ring.
(2) A ring R is UNI weakly nil-clean if and only if J(R) is nil and either R/J(R) \sim = B is a
Boolean ring, or R/J(R) \sim = \BbbZ 3, or R/J(R) \sim = B \times \BbbZ 3.
Proof. (1) \Rightarrow . With Proposition 2.3 at hand we deduce that J(R) is nil. Furthermore,
in accordance to Lemma 2.2, for any u \in U(R), we write u = n + i = 1 + n + q for some
i \in \mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R) and n, q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) such that n+ q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), because ni = in yields nq = qn.
We consequently obtain that R is nil-clean UU and so the main result in [5] applies to get the pursued
claim.
\Leftarrow . It follows from [5] that R is strongly nil-clean and hence it is UU nil-clean.
(2) \Rightarrow . Consulting with [3] (Theorem 2.17) (see also [1], Theorems 7(2) and 12(3) for the Abelian
case) one may write that R \sim = R1 \times R2, where R1 is nil-clean, and R2 is either \{ 0\} , or J(R2) is
nil and R2/J(R2) \sim = \BbbZ 3. Since R1 is nil-clean UNI, point (1) applies to get that J(R1) is nil and
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1426 P. V. DANCHEV
R1/J(R1) is Boolean. However, J(R) \sim = J(R1)\times J(R2) under the same isomorphism induced by
that of R, and hence R/J(R) \sim = [R1/J(R1)]\times [R2/J(R2)] gives the wanted claim.
\Leftarrow . It follows from [2] (see also [3]) that R is weakly nil-clean with the strong property whence
it is WUU weakly nil-clean.
Proposition 2.4 is proved.
Remark 2.4. In terms of [5], a ring R is UNI nil-clean exactly when it is UU nil-clean, that is, R
is strongly nil-clean. By the same token, in terms of [3], a ring R is UNI weakly nil-clean precisely
when it is WUU weakly nil-clean ring, that is, R is weakly nil-clean with the strong property (cf. [2]
as well).
Recall that a ring R is called local if R/J(R) is a division ring, that is, every its element lies
in either U(R) or in J(R). Note that a ring is local if and only if it is a clean ring with only trivial
idempotents, and hence it is an Abelian clean ring.
The next assertion is important for our further successful presentation.
Proposition 2.5. R is a local UNI ring if and only if J(R) is nil and either R/J(R) \sim = \BbbZ 2 or
R/J(R) \sim = \BbbZ 3.
Proof. The sufficiency being elementary, we concentrate on the necessity. Firstly, that J(R) is
nil and R/J(R) is UNI follows from Proposition 2.3.
Next, if 2 \in J(R), then Proposition 2.1 is applicable to show that the quotient R/J(R) is division
UU and thus R/J(R) \sim = \BbbZ 2, which follows either directly or in view of [5].
If now 2 \not \in J(R), then 2 \in U(R) whence by Lemma 2.1 (1) we have 3 \in J(R) and so the
factor-ring R/J(R) is UNI division of characteristic 3. Since in division rings we have no non-
trivial nilpotent elements and all involutions are only \{ - 1, 1\} , we conclude that R/J(R) \sim = \BbbZ 3, as
formulated.
The following decomposition is essential.
Lemma 2.3. A ring R is UNI for which 6 \in J(R) if and only if R \sim = R1 \times R2, where R1 is a
UU ring and R2 is either \{ 0\} or a UNI ring with 3 \in J(R2).
Proof. Necessity. Observe that for any k \in \BbbN we have (2k, 3k) = 1, i.e., there exist non-zero
integers u, v such that 2ku+ 3kv = 1. This, consequently, allows us to write that R = 2kR + 3kR.
Since 6 \in J(R), it follows in virtue of Proposition 2.3 that 6 must be nilpotent. This assures that
6m = 0 for some m \in \BbbN , and so 2mR \cap 3mR = \{ 0\} ; in fact, if x = 2ma = 3mb for some a, b \in R,
then 2mau = 3mbu. However, (1 - 3mv)a = 3mbu whence 3m(av + bu) = a. Multiplying both
sides by 2m, we derive that 0 = 2ma = x, as needed. Finally, R = 2mR \oplus 3mR and hence
R \sim = (R/2mR)\times (R/3mR) = R1 \times R2 with R1 = R/2mR \sim = 3mR and R2 = R/3mR \sim = 2mR.
Note it is not too hard to check that R1 and R2 are both UNI rings as being direct factors of a
ring isomorphic to the UNI ring R. Since 2 \in J(R1), Proposition 2.3 is a guarantor that R1/J(R1)
is UNI of characteristic 2 as well as J(R1) is nil. Thus Proposition 2.1 gives that R1/J(R1) is a UU
ring and so an application of [5] insures that R1 is UU as well. On the other hand, since 3 \in J(R2),
it follows again by Proposition 2.3 that 3 \in \mathrm{N}\mathrm{i}\mathrm{l}(R2). Notice that with the aid of Lemma 2.1 (1) we
also have 2 \in U(R2).
Sufficiency. Since 2 \in J(R1) and eventually 3 \in J(R2), provided R2 \not = \{ 0\} , it must be that
6 \in J(R), because J(R) \sim = J(R1)\times J(R2). Next, that R is UNI follows directly by observing that
the direct product of two UNI rings is again a UNI ring.
Remark 2.5. If the direct factor R2 in Lemma 2.3 is indecomposable, things will be easy. Ne-
vertheless, in contrast to WUU rings (cf. [3]), it cannot be expected that R2 will be indecomposable.
In fact, there exists a commutative UNI ring R with 2 \in U(R) such that \mathrm{I}\mathrm{d}(R) \not = \{ 0, 1\} and so
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
A GENERALIZATION OF WUU RINGS 1427
R is decomposable (compare with Example 2.1 (b) too). Indeed, consider R = \BbbZ 3 \times \BbbZ 3. Hence
2 = - 1 \in U(R). Likewise, U(\BbbZ 3) = \{ - 1, 1\} = \mathrm{I}\mathrm{n}\mathrm{v}(\BbbZ 3) with U(R) = U(\BbbZ 3) \times U(\BbbZ 3) and
\mathrm{N}\mathrm{i}\mathrm{l}(R) = \{ 0\} . Therefore, U(R) = \mathrm{I}\mathrm{n}\mathrm{v}(R) which shows that R is UNI. Moreover, it is self-evident
that R is decomposable having both (0, 1) and (1, 0) as nontrivial idempotents different to the trivial
ones (0, 0) and (1, 1).
The next statement somewhat sheds light on the complicated structure of arbitrary UNI clean
rings.
Theorem 2.1. A ring R is clean UNI if and only if R can be decomposed as the direct product of
a ring which is Boolean modulo its nil Jacobson radical and of a UNI clean ring whose nil Jacobson
radical contains 3.
In addition, if the latter ring modulo its Jacobson radical is commutative, then this factor-ring
can be embedded in the direct product of the fields \BbbZ 3.
In particular, a commutative ring R is UNI clean with 3 \in J(R) if and only if J(R) is nil and
R/J(R) \subseteq
\prod
\mu \BbbZ 3, where \mu is an ordinal.
Proof. According to Proposition 2.3, J(R) is nil. Now, knowing by Proposition 2.2 that
6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and hence 6 \in J(R), we employ Lemma 2.3 to write that R \sim = R1 \times R2, where R1 is
either \{ 0\} or is UU clean and R2 is either \{ 0\} or is UNI clean with 3 \in J(R2). As for R1, we
employ [5] to prove that R1/J(R1) is Boolean, as stated.
Concerning the other part, assume now that R2/J(R2) is commutative. Moreover, Proposition 2.3
together with [9] enable us that R2/J(R2) is also clean and UNI. Since this quotient does not contain
nontrivial nilpotents, any its element is the sum of an involution and an idempotent. Being of
characteristic 3, it easily follows that any element in it is the sum of two idempotents, because for
any element v in this factor-ring with v2 = 1 it must be that (v - 1)2 = v2 + 1 - 2v = v - 1. We
further employ [7] (see also [4]) to get the pursued claim.
Finally, it remains to show the reciprocal part that R is UNI clean, provided J(R) is nil and
R/J(R) is a subdirect product of
\prod
\mu \BbbZ 3. But one sees that R/J(R) is UNI since any unit in
\prod
\mu \BbbZ 3
has to be an involution. We henceforth observe that issue (iii) in Proposition 2.3 can be applied to
obtain that R is UNI, indeed. Moreover, since the elements x of every subring of
\prod
\mu \BbbZ 3 satisfies
the equation x3 = x which is equivalent to x = x2( - x2+x+1) where - x2+x+1 is obviously an
involution, it follows that R/J(R) is unit-regular and hence clean. Thus [9] is workable to conclude
the promised assertion.
Theorem 2.1 is proved.
In regard to the consideration above, we add the following comments: It is well known that any
Boolean ring B is a subring of the direct product of copies of the field \BbbZ 2, and vice versa. If B
is finite, then B \sim =
\prod
k \BbbZ 2 for some finite k; however, if B is infinite, analogous isomorphism is
manifestly untrue as the following classical construction shows: Let X be a set of cardinal \aleph 0 and
let R be the Boolean ring of all finite subsets of X with the usual operations given. Then R has
cardinality \aleph 0. But the direct product of \aleph 0 copies of \BbbZ 2 will have cardinality 2\aleph 0 > \aleph 0. So, any
isomorphism between them is impossible.
To finish, resuming, a commutative ring R is UNI clean if and only if R \sim = R1 \times R2, where
J(R1) is nil with R1/J(R1) \subseteq
\prod
\lambda \BbbZ 2 for some ordinal \lambda and J(R2) is nil with R2/J(R2) \subseteq
\prod
\mu \BbbZ 3
for some ordinal \mu . So, J(R) \sim = J(R1)\times J(R2) is nil and R/J(R) \sim = [R1/J(R1)]\times [R2/J(R2)] \subseteq
\subseteq
\prod
\lambda \BbbZ 2 \times
\prod
\mu \BbbZ 3.
3. Open questions. In ending, we pose the following four unsettled problems of interest:
Problem 1. Characterize exchange UNI rings. Are they clean?
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1428 P. V. DANCHEV
In that aspect classify those rings R for which J(R) is nil and R/J(R) is embedding in\prod
\lambda \BbbZ 2\times
\prod
\mu \BbbZ 3, where \lambda and \mu are some ordinals. Since a subring of such a product
\prod
\lambda \BbbZ 2\times
\prod
\mu \BbbZ 3
satisfies the equation x3 = x for all its elements x, it is a commutative von Neumman regular ring
and thus clean. We further apply [9] to infer that R is clean as well. Nevertheless, it is not at all
obvious whether R is UNI or not.
In other words, does it follow that R is UNI clean if and only if J(R) is nil and R/J(R)
is embedding in
\prod
\lambda \BbbZ 2 \times
\prod
\mu \BbbZ 3, where \lambda and \mu are some ordinals? Notice that in virtue of
Theorem 2.1, if the quotient R/J(R) is commutative, the necessity of the problem is settled in the
affirmative.
Problem 2. If R a UNI ring and e \in \mathrm{I}\mathrm{d}(R), does it follow that the corner ring eRe is also UNI?
Problem 3. Suppose that R is a ring and s \in \BbbN . Find a criterion when the full s\times s matrix ring
\BbbM s(R) is UNI.
Problem 4. Let R be a ring and G a multiplicative group. Find a necessary and sufficient
condition when the group ring R[G] is UNI.
References
1. Breaz S., Danchev P., Zhou Y. Rings in which every element is either a sum or a difference of a nilpotent and an
idempotent // J. Algebra and Appl. – 2016. – 15, № 8. – 11 p.
2. Danchev P. V. A class of weakly nil-clean rings // Irish Math. Soc. Bull. – 2016. – 77. – P. 9 – 17.
3. Danchev P. V. Weakly UU rings // Tsukuba J. Math. – 2016. – 40. – P. 101 – 118.
4. Danchev P. V. Invo-clean unital rings // Commun. Korean Math. Soc. – 2017. – 32. – P. 19 – 27.
5. Danchev P. V., Lam T. Y. Rings with unipotent units // Publ. Math. Debrecen. – 2016. – 88. – P. 449 – 466.
6. Diesl A. J. Nil clean rings // J. Algebra. – 2013. – 383. – P. 197 – 211.
7. Hirano Y., Tominaga H. Rings in which every element is the sum of two idempotents // Bull. Aust. Math. Soc. –
1988. – 37. – P. 161 – 164.
8. Lam T. Y. A first course in noncommutative rings. – Second Ed. // Grad. Texts Math. – Berlin etc.: Springer-Verlag,
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Geom. – 2013. – 30. – P. 1 – 18.
Received 17.01.16,
after revision — 14.07.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
|
| id | umjimathkievua-article-1791 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:44Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b8/c66276d8ebb8166471273a479a5f4db8.pdf |
| spelling | umjimathkievua-article-17912019-12-05T09:26:39Z A generalization of WUU rings Узагальнення WUU кiлець Danchev, P. V. Данчев, П. В. We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a common generalization of the so-called UU rings examined by Danchev – Lam in (Publ. Math. Debrecen, 2016) and of the so-called WUU rings studied by the author in (Tsukuba J. Math., 2016). Визначено клас UNI кiлець та наведено результати їх детальних дослiджень у зв’язку з чистими кiльцями, тобто основний результат статтi описує комутативнi чистi UNI кiльця з точнiстю до iзоморфiзму. Це нове поняття є типовим узагальненням так званих UU кiлець, що вивчалися Данчевим та Ламом в (Publ. Math. Debrecen, 2016), та так званих WUU кiлець, що вивчалися Данчевим в (Tsukuba J. Math., 2016). Institute of Mathematics, NAS of Ukraine 2017-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1791 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 10 (2017); 1422-1428 Український математичний журнал; Том 69 № 10 (2017); 1422-1428 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1791/773 Copyright (c) 2017 Danchev P. V. |
| spellingShingle | Danchev, P. V. Данчев, П. В. A generalization of WUU rings |
| title | A generalization of WUU rings |
| title_alt | Узагальнення WUU кiлець |
| title_full | A generalization of WUU rings |
| title_fullStr | A generalization of WUU rings |
| title_full_unstemmed | A generalization of WUU rings |
| title_short | A generalization of WUU rings |
| title_sort | generalization of wuu rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1791 |
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