Weakly clean and exchange UNI rings
We find a criterion for a commutative ring to be weakly clean UNI, as well as a criterion for an exchange ring to have strongly invo-clean unit group. These two assertions somewhat improve earlier author's results (Ukr. Math. J., 2017) and (Internat. J. Algebra, 2017).
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| author | Danchev, P. V. Данчев, П. В. Данчев, П. В. |
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| description | We find a criterion for a commutative ring to be weakly clean UNI, as well as a criterion for an exchange ring to have strongly invo-clean unit group. These two assertions somewhat improve earlier author's results (Ukr. Math. J., 2017) and (Internat. J. Algebra, 2017).
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UDC 512.5
P. V. Danchev (Inst. Math. and Inform. Bulgarian Academy of Sciences, Sofia)
WEAKLY CLEAN AND EXCHANGE UNI RINGS*
СЛАБКО ЧИСТI ТА ОБМIННI УНIКIЛЬЦЯ
We find a criterion for a commutative ring to be weakly clean UNI, as well as a criterion for an exchange ring to have
strongly invo-clean unit group. These two assertions somewhat improve earlier author’s results (Ukr. Math. J., 2017) and
(Internat. J. Algebra, 2017).
Знайдено критерiй того, що комутативне кiльце є слабко чистим унiкiльцем, а також критерiй того, що обмiнне
кiльце має сильну iнво-чисту одиничну групу. Цi два твердження дещо покращують результати автора, що були
опублiкованi ранiше в (Ukr. Math. J., 2017) та (Internat. J. Algebra, 2017).
1. Introduction and background. Everywhere in the text, all rings are assumed to be associative,
containing the identity element 1 which differs from the zero element 0. As usual, for such a ring R,
U(R) stands for the group of all units in R, \mathrm{I}\mathrm{n}\mathrm{v}(R) for its subset of all involutions, \mathrm{N}\mathrm{i}\mathrm{l}(R) for the
set of all nilpotents in R with subset \mathrm{N}\mathrm{i}\mathrm{l}2(R) consisting of all nilpotents of order less than or equal
to 2, \mathrm{I}\mathrm{d}(R) for the set of all idempotents in R, J(R) for the Jacobson radical of R, and Z(R) for
the center of R. Our further terminology and notations are in agreement with [11].
In [14] was introduced the following paramount notion.
Definition 1.1. A ring R is called clean if, for every r \in R, there exist u \in U(R) and e \in \mathrm{I}\mathrm{d}(R)
with r = u+ e.
If the unit u is replaced by the unipotent 1 + q for some q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), we obtain the proper
subclass of nil-clean rings.
This was naturally generalized in [1] to the following concept.
Definition 1.2. A ring R is called weakly clean if, for each r \in R, there exist u \in U(R) and
e \in \mathrm{I}\mathrm{d}(R) with r = u+ e or r = u - e.
If the unit u is again replaced by the unipotent 1 + q for some q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), we detect the proper
subclass of weakly nil-clean rings.
It is straightforward to see that any clean ring is weakly clean, but the converse is irreversible.
In fact, there are commutative rings which are weakly clean but not clean; e.g., such is the ring
\BbbZ (3)\cap \BbbZ (5) as well as the ring \BbbZ (15), where \BbbZ (k) =
\Bigl\{ r
s
\in \BbbQ : k \nmid s
\Bigr\}
, k \in \BbbN , is a prime or (k, s) = 1
for any other natural k. However, if 2 \in J(R), then these two classes do coincide (see [3]). Indeed,
one can represent u - e = (u - 2e) + e, where u - 2e \in U(R) + J(R) = U(R).
Definition 1.3. A ring R is called exchange if, for any r \in R, there exist e \in \mathrm{I}\mathrm{d}(rR) such that
1 - e \in (1 - r)R.
It is well known that all clean rings are exchange, while the converse fails in general. However,
for Abelian rings (i.e., for rings R with \mathrm{I}\mathrm{d}(R) \subseteq Z(R)), these two notions are equivalent.
Moreover, the weakly clean and exchange concepts are independent one to other.
On the other vein, following Cǎlugǎreanu, in [10] was given the following definition.
Definition 1.4. A ring R is said to be UU , provided that U(R) = 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R).
* This paper is partially supported by Grant KP-06 N 32/1 of 07.12.2019 of the Bulgarian National Science Fund.
c\bigcirc P. V. DANCHEV, 2019
1618 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
WEAKLY CLEAN AND EXCHANGE UNI RINGS 1619
It is clear that in these rings 2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and so 2 \in J(R).
The above property of units was generalized in [4] like this.
Definition 1.5. A ring R is said to be WUU , provided that U(R) = 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R) or U(R) =
= - 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R), i.e., U(R) = \pm 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R).
Apparently, UU rings are themselves WUU, but the converse is manifestly false by considering
rings \BbbZ and \BbbZ 3
\sim = \BbbZ /(3). Nevertheless, the WUU notion was extended in [9] as follows.
Definition 1.6. A ring R is said to be UNI , provided that U(R) = [\mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R)] + \mathrm{N}\mathrm{i}\mathrm{l}(R).
Evidently, WUU rings are themselves UNI, but the converse is untrue. However, if 2 is nilpotent,
these two concepts are tantamount, thus enlarging an assertion from [9]. In fact, for any involution
v, it must be that (1 - v)2 = 2(1 - v) whence by induction we have (1 - v)m+1 = 2m(1 - v) for
every natural number m. Consequently, 1 - v is a nilpotent and thus, for any central involution i, it
follows that 1 - i is a central nilpotent. That is why, \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R) \subseteq 1+\mathrm{N}\mathrm{i}\mathrm{l}(R)\cap Z(R) implying
that
U(R) = \mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R) + \mathrm{N}\mathrm{i}\mathrm{l}(R) \subseteq 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R) \cap Z(R) + \mathrm{N}\mathrm{i}\mathrm{l}(R) = 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R),
as required.
In [10] was found a criterion when clean ring is UU (compare with Theorem 2.1 below). Since
in UU rings the element 2 is always nilpotent, this is tantamount to a criterion when weakly clean
ring is UU. After that, in order to supersede this, in [4] and [7] was found a criterion when (weakly)
clean ring is WUU (compare with Theorem 2.2 below). These two things were partially improved in
[9] to commutative clean UNI rings and arbitrary weakly nil-clean UNI rings.
In the spirit of ring definitions from [5] and [6], we are now able to expand the UNI notions like
these.
Definition 1.7. Let R be a ring. We shall say that U(R) is invo-fine if the equality U(R) =
= \mathrm{I}\mathrm{n}\mathrm{v}(R) + \mathrm{N}\mathrm{i}\mathrm{l}(R) holds.
A little stronger condition, however, is the next one definition.
Definition 1.8. Let R be a ring. We shall say that U(R) is strongly invo-fine if, for every
u \in U(R), there are v \in \mathrm{I}\mathrm{n}\mathrm{v}(R) and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) such that u = v + q with vq = qv.
A quick look shows that UNI rings and strongly invo-fine rings have strongly invo-fine unit
groups. Also, in the commutative case UNI and invo-fine concepts are tantamount. Besides, invo-
fine rings have invo-fine unit groups. As showed in [6], all (strongly, uniquely) invo-fine rings are
just \BbbZ 2 and \BbbZ 3, but however rings with (strongly) invo-fine unit groups form a larger class which
merits a proper investigation. Notice that in the general noncommutative case Definitions 1.7 and 1.8
are independent as these two sorts of rings form wide classes (certain relevant examples are given in
[10]).
And so, the purpose of the present paper is to strengthen these achievements quoted above by
finding a necessary and sufficient condition when a commutative ring is weakly clean UNI and an
arbitrary ring is exchange with strongly invo-fine unit group. This is successfully done below in
Theorems 2.3 and 2.4, respectively. The general problems for an arbitrary weakly clean UNI ring
and for an arbitrary exchange ring with invo-fine unit group are still unsettled.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1620 P. V. DANCHEV
2. Main results. We first list the following technicality.
Lemma 2.1. In a weakly clean UNI ring, 6 or 30 is nilpotent.
Proof. In such a ring R, we write 3 = i+ n+ e or 3 = i+ n - e, where i \in \mathrm{I}\mathrm{n}\mathrm{v}(R) \cap Z(R),
n \in \mathrm{N}\mathrm{i}\mathrm{l}(R) and e \in \mathrm{I}\mathrm{d}(R). Thus, ne = en forcing that (n+e)2 - (n+e) = n2+2ne - n \in \mathrm{N}\mathrm{i}\mathrm{l}(R)
and (n - e)2 + (n - e) = n2 - 2ne + n \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Therefore, (3 - i)2 - (3 - i) \in \mathrm{N}\mathrm{i}\mathrm{l}(R) or
(3 - i)2 + (3 - i) \in \mathrm{N}\mathrm{i}\mathrm{l}(R). In the first case, one has that 7 - 5i \in \mathrm{N}\mathrm{i}\mathrm{l}(R), while in the second one
13 - 7i \in \mathrm{N}\mathrm{i}\mathrm{l}(R). So, 5i \in 7 + \mathrm{N}\mathrm{i}\mathrm{l}(R) or 7i \in 13 + \mathrm{N}\mathrm{i}\mathrm{l}(R) giving by squaring that 24 \in \mathrm{N}\mathrm{i}\mathrm{l}(R)
or 120 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Hence 63 = 9.24 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), i.e., 6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) or (30)3 = 225.120 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), i.e.,
30 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as stated.
Lemma 2.1 is proved.
Recall the following result from [10].
Theorem 2.1. Suppose that R is a ring. Then the next three points are equivalent:
(1) R is weakly clean UU ;
(2) R is clean UU ;
(3) R/J(R) is Boolean and J(R) is nil.
Recall the following result from [4, 7].
Theorem 2.2. Suppose R is a ring. Then the next three issues are equivalent:
(1) R is weakly clean WUU ;
(2) R is clean WUU ;
(3) J(R) is nil and either R/J(R) \sim = B, or R/J(R) \sim = \BbbZ 3, or R/J(R) \sim = B \times \BbbZ 3, where B is
a boolean ring.
We now come to our chief statement.
Theorem 2.3. Let R be a commutative ring. Then the next three items are equivalent:
(1) R is weakly clean UNI;
(2) R is clean UNI;
(3) J(R) is nil with R \sim = L \times P, where L/J(L) \subseteq
\prod
\lambda \BbbZ 2 and P/J(P ) \subseteq
\prod
\mu \BbbZ 3 for some
ordinals \lambda and \mu .
Proof. The equivalence (2) \Leftarrow \Rightarrow (3) follows directly from [9]. Since the implication
(2) \Rightarrow (1) is self-evident, we will show the reverse one (1) \Rightarrow (2). To that goal, we first observe
that a homomorphic image of a weakly clean ring is again weakly clean as well as a direct factor of
a UNI ring is also UNI. Furthermore, since (2, 3, 5) = 1, in view of Lemma 2.1, one can decompose
as done in [4, 9, 7], respectively, R \sim = P \times L \times T, where P,L, T are weakly UNI rings with
2 \in J(P ), 3 \in J(L) and 5 \in J(T ), provided that they are non-zero. Thus, if T \not = \{ 0\} , then
6 \in 1 + J(T ) \subseteq U(T ) yielding that both 2 \in U(T ) and 3 \in U(T ). However, in accordance with
[9], this is impossible to be fulfilled in a UNI ring, whence it must be that T = \{ 0\} .
On the other hand, exploiting [3], we derive that P is clean, because it is weakly clean for
which 2 \in J(P ). Besides, owing to [9], J(L) is nil and L/J(L) has to be weakly clean UNI
of characteristic 3. Since L/J(L) is commutative, it does not contain non-zero nilpotent elements,
that is, \mathrm{N}\mathrm{i}\mathrm{l}(L/J(L)) = \{ 0\} . So, L/J(L) = \mathrm{I}\mathrm{n}\mathrm{v}(L/J(L)) \pm \mathrm{I}\mathrm{d}(L/J(L)) writing x\prime = v\prime + e\prime
or x\prime = v\prime - e\prime for any x\prime \in L/J(L) and some v\prime \in \mathrm{I}\mathrm{n}\mathrm{v}(L/J(L)), e\prime \in \mathrm{I}\mathrm{d}(L/J(L)). Since
x\prime 3 = (v\prime + e\prime )3 = v\prime 3+ e\prime 3 = v\prime + e\prime = x\prime and x\prime 3 = (v\prime - e\prime )3 = v\prime 3 - e\prime 3 = v\prime - e\prime = x\prime , it follows
that L/J(L) is unit-regular. This means with the aid of [2] that L/J(L) is clean and thus it allows
us to infer with [14] at hand that L remains clean. That is why, R \sim = P \times L is clean as being the
direct product of two clean rings, as wanted.
Theorem 2.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
WEAKLY CLEAN AND EXCHANGE UNI RINGS 1621
A question which immediately arises is whether or not R is (weakly) clean UNI if, and only
if, J(R) is nil and R/J(R) \subseteq
\prod
\lambda \BbbZ 2 \times
\prod
\mu \BbbZ 3 for some ordinals \lambda , \mu ? We next continue with a
paralleling technical claim to that in Lemma 2.1.
Lemma 2.2. If R is a clean ring with strongly invo-clean U(R), then 6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). In particular,
R \sim = R1 \times R2, where both R1 and R2 are clean rings with strongly invo-clean unit groups U(R1)
and U(R2) such that 2 \in J(R1) and 3 \in J(R2).
Proof. Write 3 = u + e = v + q + e, where u \in U(R); v \in \mathrm{I}\mathrm{n}\mathrm{v}(R) and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) with
vq = qv; e \in \mathrm{I}\mathrm{d}(R). Therefore, it is readily seen that qe = eq and, resultantly, (3 - v)2 - (3 - v) =
= (q + e)2 - (q + e) = q2 + 2qe - q \in \mathrm{N}\mathrm{i}\mathrm{l}(R), i.e., 10 - 6v - 3 + v = 7 - 5v \in \mathrm{N}\mathrm{i}\mathrm{l}(R). This
gives 5v \in 7 + \mathrm{N}\mathrm{i}\mathrm{l}(R) and squaring the containment we derive that 24 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). This ensures that
63 = 216 = 9.24 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) forcing that 6 \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as asserted.
The second part-half is straightforward and its proof is leaved to the interested reader (for more
details, see [4, 8, 9], respectively).
Lemma 2.2 is proved.
Lemma 2.3. For every ring R \not = \{ 0\} the group U(\BbbM 2(R)) is not strongly invo-fine.
Proof. Consider the invertible matrix
\biggl(
0 - 1
- 1 1
\biggr)
with the inverse
\biggl(
- 1 - 1
- 1 0
\biggr)
. If we
assume in a way of contradiction that it is presentable as a sum of an involution matrix and a
nilpotent matrix which commute each to other, then one must have that
\biggl(
0 - 1
- 1 1
\biggr) 2
-
\biggl(
1 0
0 1
\biggr)
should be a nilpotent. But
\biggl(
0 - 1
- 1 1
\biggr) 2
=
\biggl(
1 - 1
- 1 2
\biggr)
whence
\biggl(
1 - 1
- 1 2
\biggr)
-
\biggl(
1 0
0 1
\biggr)
=
=
\biggl(
0 - 1
- 1 1
\biggr)
is a nilpotent. But as we already have just seen, this matrix is invertible, getting the
wanted contradiction.
Lemma 2.3 is proved.
The following construction is well-known and will be used in the subsequent lemma. And so, if
R is a ring with r \in R such that r2 - r \in J(R), which is a nil ideal, it follows that there is e \in \mathrm{I}\mathrm{d}(R)
with er = re and e - r \in J(R). In fact, (r2 - r)m = 0 for some m \in \BbbN , and expanding this by
the Newton’s binomial formula, we obtain that rm - rm+1
\sum m
i=1
( - 1)i - 1Cm
i ri - 1 = 0. Putting a =
=
\sum m
i=1
( - 1)i - 1Cm
i ri - 1, we conclude that rm = rm+1a and ra = ar. Taking e = (ra)m = rmam,
it follows easily that er = re and e2 = e, as promised.
We are now ready to prove the following lemma.
Lemma 2.4. Let R be a ring.
(i) If U(R) is strongly invo-fine, then J(R) is nil and U(R/J(R)) is strongly invo-fine.
(ii) If U(R/J(R)) = \mathrm{I}\mathrm{n}\mathrm{v}(R/J(R)) and J(R) is nil with 3 \in J(R), then U(R) is strongly
invo-fine.
Proof. (i) Given z \in J(R), we write that 1+z = v+q, where v \in \mathrm{I}\mathrm{n}\mathrm{v}(R) and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) with
vq = qv, because 1+J(R) \leq U(R). Squaring this equality, we find that z2+2z = 2vq+q2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R).
Replacing z by - z, we have that z2 - 2z \in \mathrm{N}\mathrm{i}\mathrm{l}(R). So, the sum (z2+2z)+(z2 - 2z) = 2z2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R)
since its members commute. Furthermore, since z2 \in J(R), adapting the same trick for this element,
we infer that z4 + 2z2 \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Thus the difference (z4 + 2z2) - 2z2 = z4 \in \mathrm{N}\mathrm{i}\mathrm{l}(R) because its
members commute. Finally, z \in \mathrm{N}\mathrm{i}\mathrm{l}(R), as required.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1622 P. V. DANCHEV
About the second part, utilizing the folklore fact that U(R) \rightarrow U(R/J(R)) is always a surjective
homomorphism, induced by the natural ring map R \rightarrow R/J(R), with kernel 1 + J(R), and using
the fact that homomorphic images of commuting involutions and nilpotents are again such elements,
we are set.
(ii) Choosing an arbitrary u \in U(R) and seeing that u+ J(R) \in U(R/J(R)) = \mathrm{I}\mathrm{n}\mathrm{v}(R/J(R)),
it follows that u2 - 1 \in J(R). This can be written as ( - u - 1)2 - ( - u - 1) = u2+2u+1+u+1 =
= u2 - 1+3u+3 \in J(R). Since J(R) is nil, with the aid of the classical construction given above,
we conclude that there is an idempotent f such that f commutes with - u - 1, and so with u, and
such that f - ( - u - 1) = f + u+ 1 \in J(R). Hence, u \in - 1 - f + J(R) and thus
u \in (2f - 1) - 3f + J(R) = 2f - 1 + J(R) \in \mathrm{I}\mathrm{n}\mathrm{v}(R) + \mathrm{N}\mathrm{i}\mathrm{l}(R).
Since u commutes with 2f - 1, it follows that 2f - 1 will commute with the existing element from
J(R) as well, and we are done.
Lemma 2.4 is proved.
We now have accumulated all the information necessary to prove our second basic assertion,
which settles Problem 1 from [9] and corresponds to Theorem 2.3 (2).
Theorem 2.4. A ring R is exchange with strongly invo-fine U(R) if and only if J(R) is nil and
R \sim = R1 \times R2, where R1/J(R1) \subseteq
\prod
\lambda \BbbZ 2 and R2/J(R2) \subseteq
\prod
\mu \BbbZ 3 for some ordinals \lambda and \mu .
Proof. \Rightarrow . First, assume J(R) = \{ 0\} and assume \mathrm{N}\mathrm{i}\mathrm{l}2(R) \not = \{ 0\} . Since R is an I-ring in the
sense of [13], [12] (Theorem 2.1) applies to find an idempotent f in R and a non-zero ring T such
that the isomorphism fRf \sim = \BbbM 2(T ) is fulfilled. Therefore, one deduces that U(fRf) \sim = U(\BbbM 2(T )).
We claim now that the square of any element of the group U(fRf) is unipotent, that is, the
sum of 1 and nilpotent. Indeed, letting u \in U(fRf) with the inverse w \in fRf, we derive that
u+(1 - f) \in U(R) with the inverse w+(1 - f). Writing u+(1 - f) = v+ q for some v \in \mathrm{I}\mathrm{n}\mathrm{v}(R)
and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R) with vq = qv, and taking into account that u(1 - f) = (1 - f)u = 0, by squaring
the above equality about u, we infer that u2+1 - f = 1+d for some d \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Hence u2 = f+d
and since u2 - f \in fRf, we conclude that d \in fRf \cap \mathrm{N}\mathrm{i}\mathrm{l}(R) = \mathrm{N}\mathrm{i}\mathrm{l}(fRf), as required.
However, the proof of Lemma 2.3 demonstrates that this is not case for the group U(\BbbM 2(T )).
Thus the above group isomorphism is impossible and, finally, this contradiction argues that \mathrm{N}\mathrm{i}\mathrm{l}2(R) =
= \{ 0\} and hence \mathrm{N}\mathrm{i}\mathrm{l}(R) = \{ 0\} . So, R being reduced yields that R is Abelian and thus, in virtue of
[14], it must be clean. Also, U(R) = \mathrm{I}\mathrm{n}\mathrm{v}(R). Consequently, R = U(R)+\mathrm{I}\mathrm{d}(R) = \mathrm{I}\mathrm{n}\mathrm{v}(R)+\mathrm{I}\mathrm{d}(R).
Now, in terms of [8], the ring R is strongly invo-clean. Thus we can now apply [8] (Corollary
2.17) accomplished with [10] (Theorem B) to get that R \sim = R1 \times R2, where R1
\sim = B, where B is
a Boolean ring, and R2 \subseteq
\prod
\mu \BbbZ 3. That is why, we may identify R \subseteq
\prod
\lambda \BbbZ 2 \times
\prod
\mu \BbbZ 3 for some
ordinals \lambda and \mu .
Furthermore, to treat the general case when J(R) \not = \{ 0\} , we detect with the aid of [14] that the
quotient R/J(R) is also exchange. Likewise, with Lemma 2.4 at hand, the ideal J(R) is nil, and
the group U(R/J(R)) \sim = U(R)/(1 + J(R)) remains strongly invo-fine. We may now employ what
we have shown above in the first part to deduce that R/J(R) is clean. Again [14] works to get that
R is clean. Now Lemma 2.2 is applicable to decompose R \sim = R1 \times R2, where R1 is clean with
2 \in J(R1) and R2 is clean with 3 \in J(R2). Since U(R) \sim = U(R1) \times U(R2), and it is not hardly
checked with the help of standard coordinate-wise arguments that a direct factor of a strongly invo-
fine group is again strongly invo-fine, we infer that both U(R1) and U(R2) are strongly invo-fine.
Thus R1/J(R1) and R2/J(R2) are both clean with strongly invo-fine unit groups. We next apply
the first part above to conclude the pursued relations.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
WEAKLY CLEAN AND EXCHANGE UNI RINGS 1623
\Leftarrow . That R is exchange follows immediately by application of [14], because both R1 and R2
are so. In fact, so are R1/J(R1) and R2/J(R2) being unit-regular rings, since their elements satisfy
the equation x3 = x, as well as J(R1) and J(R2) are nil.
On the other side, U(R1)/(1 + J(R1)) \sim = U(R1/J(R1)) = \{ 1\} and so U(R1) = 1 + J(R1) =
= 1 + \mathrm{N}\mathrm{i}\mathrm{l}(R1) implying that U(R1) is a strongly invo-fine group. Moreover, U(R2/J(R2)) =
= \mathrm{I}\mathrm{n}\mathrm{v}(R2/J(R2)) and thus Lemma 2.4 (ii) assures that U(R2) is a strongly invo-fine group, as
J(R2) is nil. Since the direct product of two strongly invo-fine groups is plainly verified to be again
a strongly invo-fine group, so will be U(R) and we are finished.
A question which immediately arises is whether or not R is exchange with strongly invo-fine
U(R) if, and only if, J(R) is nil and R/J(R) \subseteq
\prod
\lambda \BbbZ 2 \times
\prod
\mu \BbbZ 3 for some ordinals \lambda , \mu ?
A brief information concerning the inheritance of some properties by the corner subring of a
given ring is like this: If a ring R is respectively a UU ring or a WUU ring, then so is its corner
subring eRe for any e \in \mathrm{I}\mathrm{d}(R); unfortunately this is generally not the case for clean and weakly
clean rings. In this direction, the affirmation below resolves [9] (Problem 2) as well as it expands the
corresponding claims from [10, 4], respectively.
Proposition 2.1. If R is a UNI ring, then the corner ring eRe is also UNI for any e \in \mathrm{I}\mathrm{d}(R).
Proof. Given u \in U(eRe) with inverse w \in U(eRe), it plainly follows that u+(1 - e) \in U(R)
with inverse w + (1 - e) \in U(R). Thus one may write that u + (1 - e) = v + q, where v \in
\in \mathrm{I}\mathrm{n}\mathrm{v}(R)\cap Z(R) and q \in \mathrm{N}\mathrm{i}\mathrm{l}(R). Since ue = eu = u, by multiplying both sides of u+1 - e = v+q
with e on the left and right, we deduce that u = ve + qe = ev + eq. Since ve = ev, it must be
that qe = eq \in \mathrm{N}\mathrm{i}\mathrm{l}(R). But ve = ve2 = vee = eve \in eRe, so that u - ve = qe \in \mathrm{N}\mathrm{i}\mathrm{l}(eRe).
Moreover, (ve)2 = v2e2 = e and hence ve \in \mathrm{I}\mathrm{n}\mathrm{v}(eRe). Finally, because v \in Z(R), for any
r \in R we have that ve.ere = vere = evre = erve = ervee = ere.ve, and so it follows that
u = ve+ qe \in \mathrm{I}\mathrm{n}\mathrm{v}(eRe) \cap Z(eRe) + \mathrm{N}\mathrm{i}\mathrm{l}(eRe), as expected.
Proposition 2.1 is proved.
The next statement is paralleled and motivated by Lemma 2.3 alluded to above.
Proposition 2.2. Suppose R is a commutative non-zero ring. Then, for each integer n \geq 2, the
full matrix n\times n ring \BbbM n(R) is not UNI. Even more, it has unit group which is not invo-fine when
n = 2.
Proof. Since \BbbM 2(R) is isomorphic to a corner ring of \BbbM n(R), in virtue of Proposition 2.1 it
suffices to show that the ring \BbbM 2(R) is not UNI. We will prove even that its unit group is not invo-
fine. To that aim, consider the matrix
\biggl(
0 - 1
- 1 1
\biggr)
\in GL2(R). Assume in a way of contradiction
that it can be presented as the sum of an involution matrix and a nilpotent matrix. Therefore,\biggl(
0 - 1
- 1 1
\biggr)
-
\biggl(
a b
c d
\biggr)
=
\biggl(
- a - 1 - b
- 1 - c 1 - d
\biggr)
is an involution, where
\biggl(
a b
c d
\biggr)
is a nilpotent.
So,
\biggl(
- a - 1 - b
- 1 - c 1 - d
\biggr) 2
=
\biggl(
1 0
0 1
\biggr)
and, as it is well-known, a+ d is a nilpotent in R.
We next obtain the system of equations
a2 + (1 + b)(1 + c) = 1,
a(1 + b) - (1 + b)(1 - d) = 0,
a(1 + c) - (1 + c)(1 - d) = 0,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1624 P. V. DANCHEV
(1 + c)(1 + b) + (1 - d)2 = 1.
From the second equality we detect that (1+b)(a+d - 1) = 0 and from the third that (1+c)(a+
+ d - 1) = 0. As noted above, a + d is nilpotent, so that a + d - 1 has to be a unit and hence
b = c = - 1. Consequently, a2 = (1 - d)2 = 1 and d2 = 2d. Furthermore, a2 - (1 - d)2 =
= (a - 1 + d)(a + 1 - d) = 0. Taking into account that a - 1 + d is a unit, one obtains that
a+1 - d = 0, i.e., that a - d = - 1. Once again, since a+d = q is nilpotent, it follows immediately
that (a+ d) - (a - d) = q + 1 is a unit, whence 2d is a unit which insures d2 and thus d is a unit.
Finally, d2 = 2d guarantees that d = 2 and a = 1. That is why, the matrix
\biggl(
1 - 1
- 1 2
\biggr)
should be
by assumption a nilpotent, but it is not. In fact, it may easily be verified that it is a unit with the
inverse
\biggl(
2 1
1 1
\biggr)
. This contradicts our initial choice, thus giving the pursued claim after all.
Remark 2.1. Contrasting with the commutative version, it is not the case that the trace of a
nilpotent matrix over an arbitrary ring is again nilpotent. For example, let R be the quotient of
the free (noncommutative) ring on two variables x, y by the ideal generated by x2 and y2, that is,
R = \BbbZ \langle x, y\rangle /(x2, y2), and let A be the following 2\times 2 matrix over R:\biggl(
X 0
0 Y
\biggr)
,
where X = x + (x2, y2) and Y = y + (x2, y2) denote the images of x and y, respectively, in R.
Then, one checks that A2 = 0, and so A is nilpotent of index 2. But the trace X + Y of A is not a
nilpotent, since any power of X + Y is of the form (XY )m + (Y X)m or Y (XY )m +X(Y X)m,
once all terms containing X2 or Y 2 are identified with 0. Expressions of these forms are not 0 in R,
indeed, because expressions of the forms (xy)m+(yx)m and y(xy)m+x(yx)m are not elements of
the ideal (x2, y2) in the ring \BbbZ \langle x, y\rangle .
We finish off with a facilitating problem of interest (compare with Proposition 2.1 as well).
Problem. If R is a ring and e \in \mathrm{I}\mathrm{d}(R) such that U(R) is (strongly) invo-fine, is then U(eRe)
also (strongly) invo-fine?
Acknowledgments. The author is grateful to Prof. Zachary Mesyan for their valuable private
communication about the trace of arbitrary matrices.
References
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ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
WEAKLY CLEAN AND EXCHANGE UNI RINGS 1625
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Received 10.03.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
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| id | umjimathkievua-article-2288 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:51Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/42/2644c4a0d90024cddf3b6f6ca33d4542.pdf |
| spelling | umjimathkievua-article-22882020-01-29T10:46:45Z Weakly clean and exchange UNI rings Слабко чистi та обмiннi унiкiльця Danchev, P. V. Данчев, П. В. Данчев, П. В. Обмінні кільця, Чисті кільця, Слабо чисті кільця, кільця UNI, Кільця з тонкими тонками, Кільця Invo-Clean, Idempotents, Nilpotents, Involutions, Units Exchange rings, Clean rings, Weakly clean rings, UNI rings, Invo-fine rings, Invo-clean rings, Idempotents, Nilpotents, Involutions, Units We find a criterion for a commutative ring to be weakly clean UNI, as well as a criterion for an exchange ring to have strongly invo-clean unit group. These two assertions somewhat improve earlier author's results (Ukr. Math. J., 2017) and (Internat. J. Algebra, 2017). Знайдено критерій того, що комутативне кільце є слабко чистим унікільцем, а також критерій того, що обмінне кільце має сильну інво-чисту одиничну групу.Ці два твердження дещо покращують результати автора, що були опубліковані раніше в (Ukr. Math. J., 2017) та (Internat. J. Algebra, 2017). Institute of Mathematics, NAS of Ukraine 2019-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2288 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 12 (2019); 1618-1625 Український математичний журнал; Том 71 № 12 (2019); 1618-1625 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2288/1528 Copyright (c) 2019 Elina Dichter (Менеджер журналу) |
| spellingShingle | Danchev, P. V. Данчев, П. В. Данчев, П. В. Weakly clean and exchange UNI rings |
| title | Weakly clean and exchange UNI rings |
| title_alt | Слабко чистi та обмiннi унiкiльця |
| title_full | Weakly clean and exchange UNI rings |
| title_fullStr | Weakly clean and exchange UNI rings |
| title_full_unstemmed | Weakly clean and exchange UNI rings |
| title_short | Weakly clean and exchange UNI rings |
| title_sort | weakly clean and exchange uni rings |
| topic_facet | Обмінні кільця Чисті кільця Слабо чисті кільця кільця UNI Кільця з тонкими тонками Кільця Invo-Clean Idempotents Nilpotents Involutions Units Exchange rings Clean rings Weakly clean rings UNI rings Invo-fine rings Invo-clean rings Idempotents Nilpotents Involutions Units |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2288 |
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