Strongly $P$ -clean and semi-Boolean group rings
A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit. A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute. The class of stro...
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| author | Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. |
| author_facet | Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. |
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| description | A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit. A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute. The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly nil clean rings. A ring $R$ is called semi-Boolean if $R/J(R)$ is Boolean and idempotents lift modulo $J(R),$ where $J(R)$ denotes the Jacobson radical of $R.$ The class of semi-Boolean rings lies strictly between the classes of uniquely clean and clean rings. We obtain a complete characterization of strongly P-clean group rings. It is proved that the group ring $RG$ is strongly P-clean if and only if $R$ is strongly P-clean and $G$ is a locally finite 2-group. Further, we also study semi-Boolean group rings. It is proved that if a group ring $RG$ is semi-Boolean, then $R$ is a semi-Boolean ring and $G$ is a 2-group and that the converse assertion is true if $G$ is locally finite and solvable, or an FC group.
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UDC 512.5
D. Udar, R. K. Sharma, J. B. Srivastava (Indian Inst. Technology, Delhi, India)
STRONGLY \bfitP -CLEAN AND SEMI-BOOLEAN GROUP RINGS
СИЛЬНО \bfitP -ЧИСТI ТА НАПIВБУЛЕВI ГРУПОВI КIЛЬЦЯ
A ring R is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and
a unit. A ring R is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly
nilpotent element that commute. The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly
nil clean rings. A ring R is called semi-Boolean if R/J(R) is Boolean and idempotents lift modulo J(R), where J(R)
denotes the Jacobson radical of R. The class of semi-Boolean rings lies strictly between the classes of uniquely clean and
clean rings. We obtain a complete characterization of strongly P-clean group rings. It is proved that the group ring RG is
strongly P-clean if and only if R is strongly P-clean and G is a locally finite 2-group. Further, we also study semi-Boolean
group rings. It is proved that if a group ring RG is semi-Boolean, then R is a semi-Boolean ring and G is a 2-group and
that the converse assertion is true if G is locally finite and solvable, or an FC group.
Кiльце R називається чистим (вiдповiдно, однозначно чистим), якщо кожний його елемент допускає (однозначне)
зображення у виглядi суми iдемпотента та одиницi. Кiльце R називається сильно P-чистим, якщо кожний його
елемент допускає зображення у виглядi суми iдемпотента та сильно нiльпотентного елемента, що комутують. Клас
сильно P-чистих кiлець є пiдкласом класiв напiвбулевих та сильно нульових чистих кiлець. Кiльце R називається
напiвбулевим, якщо R/J(R) є булевим, а iдемпотенти пiднiмають по модулю J(R), де J(R) — радикал Джекобсона
для R. Клас напiвбулевих кiлець лежить точно мiж класами однозначно чистих та чистих кiлець. Отримано повну
характеризацiю сильно P-чистих групових кiлець. Доведено, що групове кiльце RG є сильно P-чистим тодi i тiльки
тодi, коли R є сильно P-чистим, а G — локально скiнченна 2-група. Крiм того, вивчаються також напiвбулевi
груповi кiльця. Доведено, що у випадку, коли групове кiльце RG є напiвбулевим, R — напiвбулевим кiльцем, а G —
2-групою, обернене твердження є справедливим, якщо G є локально скiнченною та розв’язною або ж FC-групою.
1. Introduction. Throughout this paper, R is an associative ring with identity 1 \not = 0. Further, J(R)
and P (R) denote the Jacobson and Prime radical of R, respectively. A ring R is said to be clean
if every element is sum of an idempotent and a unit; uniquely clean if every element is uniquely the
sum of an idempotent and a unit. A ring R is strongly clean if every element is sum of an idempotent
and a unit that commute with each other; nil clean if every element is sum of an idempotent and a
nilpotent element. A ring R is strongly P-clean if every element x \in R can be written as x = e+ b,
e2 = e, b \in P (R) and eb = be. A ring R is called semi-Boolean if R/J(R) is Boolean and
idempotents lift modulo J(R). Clean rings were introduced by Nicholson [9]. Noncommutative
uniquely clean rings were studied by Nicholson and Zhou [10]. Nil clean rings were studied by Diesl
[4]. Strongly P-clean rings were introduced by Chen, Köse and Kurtulmaz [1]. Semi-Boolean rings
were introduced by Nicholson and Zhou [11]. They show that the class of semi-Boolean rings lies
strictly between the class of uniquely clean rings and the class of clean rings:
uniquely clean \Rightarrow semi-Boolean \Rightarrow clean.
None of the above implications are reversible. The ring of upper triangular matrices Tn(\BbbZ 2) over the
field of two elements \BbbZ 2, is not uniquely clean though it is semi-Boolean [11]. Also, \BbbZ 9, the ring of
integers modulo 9, is clean but it is not semi-Boolean [11].
Group ring of a group G and a ring R is denoted by RG. The augmentation map \omega : RG \rightarrow R
is defined by \omega
\Bigl( \sum
g\in G
rgg
\Bigr)
=
\sum
g\in G
rg. If H is a subgroup of G, then \omega H will denote the right
ideal of RG generated by \{ 1 - h| h \in H\} . It is easy to see that \omega H is a two sided ideal of RG if
c\bigcirc D. UDAR, R. K. SHARMA, J. B. SRIVASTAVA, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12 1717
1718 D. UDAR, R. K. SHARMA, J. B. SRIVASTAVA
H is a normal subgroup of G. In particular, RG/\omega H \sim = R(G/H) and hence RG/\omega G \sim = R. If I is
an ideal of R, then IG is an ideal of RG and RG/IG \sim = (R/I)G. For related results, we refer to
Connell [3] and Passman [12].
The FC-center \Delta (G) of a group G is the set of all elements of G which have finitely many
conjugates in G. It is easy to see that \Delta (G) = \{ x \in G| | G : CG(x)| < \infty \} , and \Delta +(G) = \{ x \in
\in G| | G : CG(x)| < \infty and o(x) < \infty \} . If G = \Delta +(G), then G is locally normal, i.e., every finite
subset of G is contained in a finite normal subgroup of G.
In Section 2, we obtain a complete characterization of strongly P-clean group rings. It is proved
that RG is strongly P-clean if and only if R is strongly P-clean and G is a locally finite 2-group.
In Section 3, we determine when is a group ring RG semi-Boolean? It is proved that if RG is
semi-Boolean, then R is semi-Boolean and G is a 2-group. This result is a generalization of [2]
(Theorem 5). The converse holds, if G is a locally finite, solvable or an FC group. In case, when
RG is a commutative semi-Boolean ring, then we show that RG is semi-Boolean if and only if R is
semi-Boolean and G is a 2-group. This result is a generalization of [7] (Theorem 2.6).
2. Strongly P-clean group rings. In this section we completely characterize strongly P-clean
group rings. It is proved that the group ring RG is strongly P-clean if and only if R is strongly
P-clean, and G is a locally finite 2-group.
Recall that an ideal I of a ring R is locally nilpotent, if every finitely generated subring of I is
nilpotent. An element a \in R is strongly nilpotent, if every sequence a = a0, a1, a2, . . . such that
ai+1 \in aiRai terminates to zero. The prime radical, P (R) of a ring R consists of precisely the
strongly nilpotent elements [6, p. 170] (Ex. 10.17). Thus P (R) is a nil ideal and P (R) \subseteq J(R).
Before we prove the main result of this section, we list some of the preliminary results about strongly
P-clean rings from Chen, Köse and Kurtulmaz [1] as in the following lemma.
Lemma 2.1. (1) A ring R is strongly P-clean if and only if R is strongly clean, R/J(R) is
Boolean and J(R) is locally nilpotent.
(2) A ring R is strongly P-clean if and only if R/P (R) is Boolean.
(3) Every homomorphic image of a strongly P-clean ring is strongly P-clean.
(4) Let I be a nilpotent ideal of a ring R. Then R is strongly P-clean if and only if R/I is
strongly P-clean.
For a commutative ring, the property of being strongly P-clean and nil clean are equivalent.
Lemma 2.2. Let R be a commutative ring. Then R is strongly P-clean ring if and only if R is
nil clean
Proof. Obviously if R is strongly P-clean, then R is nil clean.
Conversely, let R be a nil clean ring. Then R/J(R) is Boolean and J(R) is nil by [4] (Corol-
lary 3.20). Since, R is commutative, an element a \in R is nilpotent if and only if it is strongly
nilpotent. Also as J(R) is nil, so it follows that J(R) = P (R). Thus R/P (R) = R/J(R) is
Boolean. Hence, by Lemma 2.1(2), R is strongly P-clean.
Now if R is strongly P-clean, then R/J(R) is Boolean and J(R) is locally nilpotent (Lem-
ma 2.1(1)). Since idempotents can be lifted modulo any nil ideal I of R. Therefore, every strongly
P-clean ring is semi-Boolean. The following example shows that the converse is not true.
Example 2.1. Let R = \BbbZ 2 \times \BbbZ 4 \times \BbbZ 8 . . . . Then R is not nil clean since the element r =
= (0, 2, 2, 2, . . .) \in R is not nil clean. Since R is commutative, by Lemma 2.2, R can not be
strongly P-clean. However, since \BbbZ 2,\BbbZ 4,\BbbZ 8, . . . are local rings with \BbbZ 2/J(\BbbZ 2) \sim = \BbbZ 2,\BbbZ 4/J(\BbbZ 4) \sim =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
STRONGLY P -CLEAN AND SEMI-BOOLEAN GROUP RINGS 1719
\sim = \BbbZ 2,\BbbZ 8/J(\BbbZ 8) \sim = \BbbZ 2, . . . , we get \BbbZ 2,\BbbZ 4,\BbbZ 8, . . . to be semi-Boolean rings. Hence, R is semi-
Boolean [11] (Example 25(3)).
Lemma 2.3. If R is a boolean ring and G is a locally finite 2-group, then RG is strongly
P-clean.
Proof. Suppose R is boolean and G is a locally finite 2-group. By [2] (Lemma 9), RG is
uniquely clean. Thus RG/J(RG) is Boolean by [10] (Theorem 20) and RG is strongly clean by
[10] (Lemma 4). Since R is Boolean, J(R) = 0 and 2 = 0 in R. We get J(RG) = \omega G by [3]
(Proposition 16(iv)). Further, by [3, p. 682] (Corollary), \omega G is locally nilpotent. Hence, J(RG) is
locally nilpotent. So it follows from Lemma 2.1(1) that RG is strongly P-clean.
We now prove the main theorem of this section.
Theorem 2.1. The group ring RG is strongly P-clean if and only if R is strongly P-clean and
G is a locally finite 2-group.
Proof. Let RG be strongly P-clean. Since the augmentation map \omega : RG \rightarrow R is an epimor-
phism, R is strongly P-clean (Lemma 2.1(3)). Thus R/P (R) is Boolean (Lemma 2.1(2)). Hence \BbbZ 2
is an image of R, whence \BbbZ 2G is strongly P-clean (image of RG). So \BbbZ 2G/P (\BbbZ 2G) is Boolean.
We have 1 - g \in \BbbZ 2G/P (\BbbZ 2G) for all g \in G. As \BbbZ 2G/P (\BbbZ 2G) is Boolean; so 1 - g = 1 - g =
= 1 - 1 = 0. Thus 1 - g \in P (\BbbZ 2G) for all g \in G. And hence \omega G \subseteq P (\BbbZ 2G) \subseteq J(\BbbZ 2G). Since
\BbbZ 2G/\omega G \sim = \BbbZ 2, we get \omega G to be maximal and \omega G = P (\BbbZ 2G) = J(\BbbZ 2G). Now by Lemma 2.1(1),
\omega G is locally nilpotent. It follows from [3, p. 682] (Corollary), that G is a locally finite 2-group.
Conversely, suppose R is a strongly P-clean ring and G is a locally finite 2-group. By [3]
(Proposition 9), we get P (R)G \subseteq P (RG). Further, P (RG/P (R)G) = P (RG)/P (R)G. Thus
RG
P (RG)
\sim =
RG/P (R)G
P (RG)/P (R)G
=
RG/P (R)G
P (RG/P (R)G)
\sim =
RG
P (RG)
where R = R/P (R). Next, R is Boolean, so by Lemma 2.3, RG is strongly P-clean. Thus
RG/P (RG) is Boolean, and hence RG/P (RG) is Bhoolean. Now it follows from Lemma 2.1(2)
that RG is strongly P-clean.
Theorem 2.1 is proved.
3. Semi-Boolean group rings. First of all we list few of the preliminary results about semi-
Boolean rings from Nicholson and Zhou [11].
Lemma 3.1. (1) A ring R is semi-Boolean if and only if R/J(R) is Boolean and idempo-
tents lift modulo J(R).
(2) A ring R is local and semi-Boolean if and only if R/J(R) \sim = \BbbZ 2.
(3) Every homomorphic image of a semi-Boolean ring is again semi-Boolean.
(4) A direct product
\prod
iRi or a direct sum
\bigoplus
iRi of rings is semi-Boolean if and only if each
Ri is semi-Boolean.
(5) If n \geq 1, then Tn(R) is semi-Boolean if and only if R is semi-Boolean.
First we make an observation that if the coefficient ring R is semi-Boolean, then RG may not
be semi-Boolean even if G is finite.
Example 3.1. Let \BbbZ (2) denotes the localization of \BbbZ at the prime ideal generated by 2 and C7 be
a cyclic group of order 7. The ring \BbbZ (2) is local with \BbbZ (2)/J(\BbbZ (2)) \sim = \BbbZ 2. Thus \BbbZ (2) is semi-Boolean.
But \BbbZ (2)C7 is not clean (by [13], Remark 18). Hence, \BbbZ (2)C7 is not semi-Boolean.
We obtain a necessary condition for the group ring RG to be semi-Boolean.
Theorem 3.1. If RG is semi-Boolean, then R is semi-Boolean and G is a 2-group.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1720 D. UDAR, R. K. SHARMA, J. B. SRIVASTAVA
Proof. Clearly, the augmentation map \omega : RG \rightarrow R is an epimorphism, and thus R is semi-
Boolean by Lemma 3.1(3). Hence R/J(R) is Boolean. Thus \BbbZ 2 is an image of R/J(R). Hence,
\BbbZ 2 is an image of R, and so \BbbZ 2G is an image of RG. Therefore, \BbbZ 2G is semi-Boolean and
\BbbZ 2G/J(\BbbZ 2G) is Boolean. Thus \alpha - \alpha 2 \in J(\BbbZ 2G) for all \alpha \in \BbbZ 2G. In particular, g \in \BbbZ 2G for all
g \in G. But then
1 - g = g - 1(g - g2) \in J(\BbbZ 2G)
for all g \in G. This proves that \omega G \subseteq J(\BbbZ 2G). By [3] (Proposition 15(i)), G is a 2-group.
Theorem 3.1 is proved.
The converse of the above theorem is true if G is locally finite. But, before that we prove a
lemma which we will need.
Lemma 3.2. A ring R is semi-Boolean if and only if R is clean and R/J(R) is Boolean.
Proof. (\Rightarrow ) Every semi-Boolean ring is clean, and if R is semi-Boolean, then R/J(R) is
Boolean.
(\Leftarrow ) Suppose R is a clean ring and R/J(R) is Boolean. Since every clean ring is an exchange
ring. Thus idempotents lift modulo J(R). Also since R/J(R) is Boolean, R is semi-Boolean (by
Lemma 3.1(1)).
Theorem 3.2. If R is a ring and G is locally finite, then RG is semi-Boolean if and only if R
is semi-Boolean and G is a 2-group.
Proof. If RG is semi-Boolean, the result follows from Theorem 3.1.
Conversely, suppose R is a semi-Boolean ring and G is a 2-group. We have R/J(R) is Boolean,
and so 2 \in J(R). Thus it follows from [14] (Theorem 4) that RG is clean. Now by [2] (Lemma 9),
(R/J(R))G \sim = RG/J(R)G is uniquely clean. Since G is locally finite, it follows from [3] (Propo-
sition 9) that J(R)G \subseteq J(RG). We now consider the map \phi : RG/J(R)G \rightarrow RG/J(RG) defined
as
\phi (\alpha + J(R)G) = \alpha + J(RG), \alpha \in RG.
It is easy to see that \phi is a ring epimorphism. So RG/J(RG) is uniquely clean. Thus
RG/J(RG)
J(RG/J(RG))
\sim = RG/J(RG)
is Boolean. Hence, it follows from Lemma 3.2 that RG is semi-Boolean.
Corollary 3.1. If G is a solvable or an FC group, then RG is semi-Boolean if and only if R is
semi-Boolean and G is a 2-group.
Proof. It is easy to see that a torsion solvable group is locally finite. Also, if G is a torsion
FC group, then G = \Delta +(G). It is known that \Delta +(G) is locally finite. The result follows from
Theorem 3.2.
Theorem 3.3. Let R be an Artinian ring, then RG is semi-Boolean iff RG is clean and
(R/J(R))G is semi-Boolean.
Proof. One way is straight forward as every semi-Boolean ring is clean and (R/J(R))G is
semi-Boolean (image of RG).
Conversely, suppose (R/J(R))G is semi-Boolean. Since R is Artinian, J(R)G \subseteq J(RG) (by
[3], Proposition 9). Now consider the map \phi : RG/J(R)G \rightarrow RG/J(RG) defined as
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
STRONGLY P -CLEAN AND SEMI-BOOLEAN GROUP RINGS 1721
\phi (\alpha + J(R)G) = \alpha + J(RG), \alpha \in RG.
Following the proof of Theorem 3.2, we get that RG/J(RG) is Boolean. Also since RG is clean,
RG is semi-Boolean (by Lemma 3.2).
Theorem 3.3 is proved.
The following result about strongly nil clean rings is proved by Koşan, Wang and Zhou [5].
Lemma 3.3 ([5], Theorem 2.7). A ring R is strongly nil clean if and only if R/J(R) is Boolean
and J(R) is nil.
It is well known that if I is any nil ideal of a ring R, then idempotents modulo I can be lifted
to R. Thus every strongly nil clean ring is semi-Boolean. So the class of strongly nil clean rings is
contained in the class of semi-Boolean rings.
Remark 3.1. A semi-Boolean ring may not be strongly nil clean ring. The ring R = \BbbZ 2 \times \BbbZ 4 \times
\times \BbbZ 8, . . . , of Example 2.1, is semi-Boolean, but not strongly nil clean.
The following characterization for a commutative group ring to be nil clean is due to McGovern,
Raja and Sharp [7].
Lemma 3.4 ([7], Theorem 2.6). Suppose R is a commutative ring and G is an Abelian group.
The group ring RG is nil clean if and only if R is nil clean and G is a 2-group.
Since every Abelian group is an FC group, Corollary 3.1 gives a generalization of the above
result. We state this as the following corollary.
Corollary 3.2. Let R be a commutative ring and G be an Abelian group. Then the group ring
RG is semi-Boolean if and only if R is semi-Boolean and G is a 2-group.
Theorem 3.4. Let R be a local ring and G be a locally finite group, then the following are
equivalent:
(1) Tn(RG) is semi-Boolean,
(2) RG/J(RG) \sim = \BbbZ 2,
(3) RG is uniquely clean,
(4) R/J(R) \sim = \BbbZ 2 and \BbbZ 2G is semi-Boolean.
Proof. 1) \Rightarrow 2) Suppose Tn(RG) is semi-Boolean, then by Lemma 3.1(5), RG is semi-Boolean.
So by Theorem 3.1, R is semi-Boolean and G is a 2-group. Also since R is local, R/J(R) \sim = \BbbZ 2.
Thus by [8] (Theorem), RG is local. Now RG is semi-Boolean and local. Hence, RG/J(RG) \sim = \BbbZ 2.
2) \Rightarrow 3) Follows from [10] (Theorem 15).
3) \Rightarrow 4) Let RG be uniquely clean, then R is uniquely clean and R/J(R) is Boolean. Also
since R is local, R \sim = \BbbZ 2. Thus \BbbZ 2 is an image of R. So \BbbZ 2G is an image of RG. Hence, \BbbZ 2G is
uniquely clean. Since a uniquely clean ring is semi-Boolean, \BbbZ 2G is semi-Boolean.
4) \Rightarrow 1) If R/J(R) \sim = \BbbZ 2, then R is semi-Boolean (by [11], Proposition 31). Also if \BbbZ 2G is
semi-Boolean, then, by Theorem 3.1, G is a 2-group. So, by Theorem 3.2, RG is semi-Boolean.
Hence, by Lemma 3.1(5), Tn(RG) is semi-Boolean.
Theorem 3.4 is proved.
References
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5. Koşan T., Wang Z., Zhou Y. Nil-clean and strongly nil-clean rings // J. Pure and Appl. Algebra. – 2016. – 220, № 2. –
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Received 07.06.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
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| last_indexed | 2026-03-24T02:21:53Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/10/65775378371c5cdf0ca6ef5823037b10.pdf |
| spelling | umjimathkievua-article-22892020-01-17T12:46:34Z Strongly $P$ -clean and semi-Boolean group rings Сильно $P$ -чистi та напiвбулевi груповi кiльця Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Strongly P-clean rings, semiboolean rings, group rings, clean rings A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit. A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute. The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly nil clean rings. A ring $R$ is called semi-Boolean if $R/J(R)$ is Boolean and idempotents lift modulo $J(R),$ where $J(R)$ denotes the Jacobson radical of $R.$ The class of semi-Boolean rings lies strictly between the classes of uniquely clean and clean rings. We obtain a complete characterization of strongly P-clean group rings. It is proved that the group ring $RG$ is strongly P-clean if and only if $R$ is strongly P-clean and $G$ is a locally finite 2-group. Further, we also study semi-Boolean group rings. It is proved that if a group ring $RG$ is semi-Boolean, then $R$ is a semi-Boolean ring and $G$ is a 2-group and that the converse assertion is true if $G$ is locally finite and solvable, or an FC group. Кільце $R$ називається чистим (відповідно, однозначно чистим), якщо кожний його елемент допускає (однозначне) зображення у вигляді суми ідемпотента та одиниці. Кільце $R$ називається сильно P-чистим, якщо кожний його елемент допускає зображення у вигляді суми ідемпотента та сильно нільпотентного елемента, що комутують.&nbsp;Клас сильно P-чистих кілець є підкласом класів напівбулевих та сильно нульових чистих кілець.&nbsp;Кільце $R$ називається напівбулевим, якщо $R/J(R)$ є булевим, а ідемпотенти піднімають по модулю $J(R),$ де $J(R)$~--- радикал Джекобсона для $R.$Клас напівбулевих кілець лежить точно між класами однозначно чистих та чистих кілець.Отримано повну характеризацію сильно P-чистих групових кілець.&nbsp;Доведено, що групове кільце $RG$ є сильно P-чистим тоді і тільки тоді, коли $R$ є сильно P-чистим, а $G$~--- локально скінченна 2-група.Крім того, вивчаються також напівбулеві групові кільця.&nbsp;Доведено, що у випадку, коли групове кільце $RG$ є напівбулевим, $R$ --- напівбулевим кільцем, а $G$ ---&nbsp;2-групою, обернене твердження є справедливим, якщо $G$ є локально скінченною та розв'язною або ж FC-групою. Institute of Mathematics, NAS of Ukraine 2020-01-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2289 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 12 (2019); 1717-1722 Український математичний журнал; Том 71 № 12 (2019); 1717-1722 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/2289/1533 Copyright (c) 2020 Д. Удар,Р. К. Шарма,Я. Б. Шривастава |
| spellingShingle | Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Udar, D. Sharma, R. K. Srivastava, J. B. Strongly $P$ -clean and semi-Boolean group rings |
| title | Strongly $P$ -clean and semi-Boolean group rings |
| title_alt | Сильно $P$ -чистi та напiвбулевi груповi кiльця |
| title_full | Strongly $P$ -clean and semi-Boolean group rings |
| title_fullStr | Strongly $P$ -clean and semi-Boolean group rings |
| title_full_unstemmed | Strongly $P$ -clean and semi-Boolean group rings |
| title_short | Strongly $P$ -clean and semi-Boolean group rings |
| title_sort | strongly $p$ -clean and semi-boolean group rings |
| topic_facet | Strongly P-clean rings semiboolean rings group rings clean rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2289 |
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