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44URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 48-52
DOI: 10.15826/umj.2019.1.005
COMMUTATIVE WEAKLY INVO-CLEAN GROUP RINGS
Peter V. Danchev
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria, danchev@math.bas.bg; pvdanchev@yahoo.com
Abstract: A ring R is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring R and each abelian group G, we find only in terms of R, G and their sections a necessary and sufficient condition when the group ring R[G] is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings.
Keywords: Invo-clean rings, Weakly invo-clean rings, Group rings.
Introduction and conventions
Throughout the current paper, we shall assume that all rings R are associative, containing the identity element 1 which differs from the zero element 0. Our standard terminology and notation are in agreement with [9] and [10], while the specific notions and notations will be stated explicitly below. As usual, J(R) denotes the Jacobson radical of a ring R and G is a multiplicative group. Both objects R and G forming the symbol R[G] will stand for the group ring of G over R.
The next concept appeared in [1], [2] and [3], respectively.
Definition 1. A ring R is said to be invo-clean if, for every r € R, there exist an involution v and an idempotent e such that r = v + e. If r = v + e or r = v — e, the ring is called weakly invo-clean.
The next necessary and sufficient condition for a commutative ring R to be invo-clean was established in [1, 2], namely: A ring R is invo-clean if, and only if, R = R1 x R2, where R1 is a nil-clean ring with z2 = 2z for all z € J(R1), and R2 is a ring of characteristic 3 whose elements satisfy the equation x3 = x.
Let us recall that a ring is nil-clean if every its element is a sum of a nilpotent and an idempotent, and it is weakly nil-clean if every its element is a sum or a difference of a nilpotent and an idempotent (see, for more details, [6]).
A criterion for an arbitrary commutative group ring to be nil-clean was recently obtained in [8]. Specifically, the following holds: A commutative ring R[G] is nil-clean if, and only if, the ring R is nil-clean and the group G is a 2-group. This was generalized in [6, Theorem 2.1] by finding a suitable criterion when R[G] is weakly nil-clean.
Some other related results in this subject can be found by the interested reader in [4] too.
So, the aim of this brief article is to obtain a paralleling result for the class of weakly invo-clean rings. This is successfully done below in our main Theorem 1.
1. The characterization result and a problem
We begin here with the following key formula from [7]: Suppose that R is a commutative ring and G is an abelian group. Then
J(R[G]) = J(R)[G] + (r(g - 1) | g € Gp, pr € J(R)>,
where Gp designates the p-primary component of G.
The next technicality already was mentioned above, but for the sake of completeness and reader's convenience, we will state it once again.
Lemma 1. [1,2] Let R be a commutative ring. Then the following two points hold:
(i) If 2 € J(R), then R is invo-clean ^^ R is nil-clean and z2 = 2z for each z € J(R).
(ii) If char(R) = 3, then R is invo-clean ^^ x3 = x for all x € R. We also need the following two technical claims.
Lemma 2. The direct product K x L of two rings K, L is invo-clean ^^ both K and L are invo-clean rings.
Proof. It is straightforward by using of results from [1] and [2]. □
Lemma 3. A ring R is weakly invo-clean ^^ either R is invo-clean or R can be decomposed as R = K x Z5, where K = {0} or K is invo-clean.
Proof. It is straightforward by the utilization of results from [2] and [3]. □
We are now ready to proceed by proving the following preliminary statement (see [5] as well).
Proposition 1. Suppose R is a non-zero commutative ring and G is an abelian group. Then R[G] is invo-clean if, and only if, R is invo-clean having the decomposition R = Ri x R2 such that precisely one of the next three items holds:
(0) G = {1}
or
(1) |G| > 2, G2 = {1}, Ri = {0} or Ri is a ring of char(Ri) = 2, and R2 = {0} or R2 is a ring of char(R2) = 3
or
(2) |G| = 2, 2r2 = 2r1 for all r1 € R1 (in addition 4 = 0 in R1), and R2 = {0} or R2 is a ring of char(R2) = 3.
P r o o f. If G is the trivial identity group, there is nothing to do, so we shall assume hereafter that G is non-identity.
"Necessity." Since there is an epimorphism R[G] ^ R, and an epimorphic image of an invo-clean ring is obviously an invo-clean ring (see, e.g., [1]), it follows at once that R is again an invo-clean ring. According to the criterion for invo-cleanness alluded to above, one writes that R = R1 x R2, where R1 is a nil-clean ring with a2 = 2a for all a € J(R1) and R2 is a ring whose
elements satisfy the equation x3 = x. Therefore, it must be that R[G] = R1[G] x R2[G], where it is not too hard to verify by Lemma 2 that both R1[G] and R2[G] are invo-clean rings.
First, we shall deal with the second direct factor R2[G] being invo-clean. Since char(R2) = 3, it follows immediately that char(R2[G]) = 3 too. Thus an application of Lemma 1 (ii) (which is an assemble of facts from [1, 2]) allows us to deduce that all elements in R2[G] also satisfy the equation y3 = y. So, given g € G C R[G], it follows that g3 = g, that is, g2 = 1.
Next, we shall treat the invo-cleanness of the group ring R1[G]. Since char(R1) is a power of 2 (see [1]), it follows the same for R1[G]. Consequently, utilizing once again Lemma 1 (i) (being an assortment of results from [1, 2]), we infer that R1[G] should be nil-clean, so that z2 = 2z for all z € J(R1[G]). That is why, invoking the criterion from [8], listed above, we have that G is a 2-group. We claim that even G2 = 1. In fact, for an arbitrary g € G, we derive with the aid of the aforementioned formula from [7] that 1 — g € J(R1 [G]), because 2 € J(R1). Hence (1 — g)2 = 2(1 — g) which forces that 1 — 2g + g2 =2 — 2g and that g2 = 1, as desired. We now assert that char(R1) = 2 whenever |G| > 2. To that purpose, there are two nonidentity elements g = h in G with g2 = h2 = 1. Furthermore, again appealing to the formula from [7], the element 1 — g + 1 — h = 2 — g — h lies in J№[G]), because 2 € J(R1). Thus (2 — g — h)2 = 2(2 — g — h) which yields that 2 — 2g — 2h + 2gh = 0. Since gh = 1 as for otherwise g = h-1 = h, a contradiction, this record is in canonical form. This assures that 2 = 0, as wanted.
However, in the case when |G| = 2, i.e. when G = {1,g | g2 = 1} = (g), we can conclude that 2r2 = 2r for any r € R1. Indeed, in view of the already cited formula from [7], the element r(1 — g) will always lie in J(R1[G]), because 2 € J(R1). We therefore may write [r(1 — g)]2 = 2r(1 — g) which ensures that 2r2 — 2r2g = 2r — 2rg is canonically written on both sides. But this means that 2r2 = 2r, as pursued. Substituting r = 2, one obtains that 4 = 0. Notice also that 2r2 = 2r for all r € R1 and a2 = 2a for all a € J(R1) will imply that a2 = 0.
"Sufficiency." Foremost, assume that (1) is true. Since R1 has characteristic 2, whence it is nil-clean, and G is a 2-group, an appeal to [8] allows us to get that R1[G] is nil-clean as well. Since z2 = 2z = 0 for every z € J(R1), it is routinely checked that 52 = 25 = 0 for each 5 € J(R1[G]), exploiting the formula from [7] for J(R1[G]) and the fact that R1[G] is a modular group algebra of characteristic 2. That is why, by a consultation with Lemma 1 (i), one concludes that R1[G] is invo-clean, as expected. Further, by the usage of Lemma 1 (ii) above, we derive that R2[G] is an invo-clean ring of characteristic 3. To see that, given x € R2[G], we write x = ^geG rgg with rg € R2 satisfying r3 = rg. Since G2 = 1 will easily imply that g3 = g, one obtains that
= E rgg)3 = £ rgg3 = £ rgg = x'
g€G geG geG
as needed. We finally conclude with the help of Lemma 2 that R[G] = R1[G] x R2[G] is invo-clean, as expected.
Let us now point (2) be fulfilled. Since G2 = 1, similarly to (1), R2 being invo-clean of characteristic 3 implies that R2[G] is invo-clean, too. In order to prove that R1 [G] is invo-clean, we observe that R1 is nil-clean with 2 € J(R1). According to [8], the group ring R1[G] is also nil-clean. What remains to show is that for any element 5 of J(R1[G]) the equality 52 = 25 is valid. Since in conjunction with the explicit formula quoted above for the Jacobson radical, an arbitrary element in J(R1[G]) has the form j + j'g + r(1 — g), where j, j' € J(R1) and r € R1, we have that [j + j'g + r(1 — g)]2 € (J(R1 )2 + 2J(R1 ))[G] + r2(1 — g)2. However, using the given conditions, z2 = 2z = 2z2 and thus z2 = 2z = 0 for any z € J(R1). Consequently, one checks that [j + j'g + r(1 — g)]2 = r2(1 — g)2 = 2r2(1 — g) = 2r(1 — g) = 2[j + j'g + r(1 — g)], because 2r2 = 2r, as required. Therefore, R1[G] is invo-clean with Lemma 1 (i) at hand. Finally, Lemma 2 gives that R[G] = R1[G] x R2[G] is invo-clean, as promised. □
It is worthwhile noticing that concrete examples of an invo-clean ring of characteristic 4, such that its elements are solutions of the equation 2r2 = 2r, are the rings Z4 and Z4 x Z4.
We will prove now the following reduction of weak invo-cleanness.
Proposition 2. Suppose that R is a commutative non-zero ring and G is an abelian group. Then R[G] is weakly invo-clean which is not invo-clean if, and only if, R is a weakly invo-clean ring which is not invo-clean and G = {1}.
Proof. "Necessity." As it is well known and easy to establish that there is a surjection R[G] ^ R, we may apply [2] to get that R is weakly invo-clean as well. According now to Lemma 3 we obtain that R is either invo-clean, or isomorphic to Z5, or decomposed as K x Z5, where K is non-zero invo-clean. We will consider these three possibilities separately:
Case 1: R is invo-clean. Since both R[G] and R have equal characteristics, it follows once again with the aid of Lemma 3 that R[G] must be invo-clean too, a contrary to our assumption.
Case 2: R = Z5. It follows that R[G] = Z5[G] has to be weakly invo-clean of characteristic 5. Employing [2], one infers that Z5[G] = Z5 whence these two rings have equal cardinalities. This, however, implies by a simple comparison of elements that G = {1}.
Case 3: R = K x Z5 with K = {0} invo-clean. Hence R[G] = K[G] x Z5G It follows as is Case 1 that K[G] is necessarily invo-clean, whereas Z5[G] is weakly invo-clean. Similarly to Case 2, we detect once again that G = {1}.
"Sufficiency." It is immediate, because of the fulfillment of the isomorphism R[G] = R. □
So, combining both Propositions 1 and 2, we come to our chief result. Specifically, the following assertion is true:
Theorem 1. Let G be an abelian group and let R be a commutative non-zero ring. Then the group ring R[G] is weakly invo-clean if, and only if, at most one of the following points is true:
(1) G = {1} and R is weakly invo-clean.
(2) G = {1} and R = R1 x R2 is invo-clean such that either
(2.1) |G| > 2, G2 = {1}, R1 = {0} or R1 is a ring of char(R1) = 2, and R2 = {0} or R2 is a ring of char(R2) = 3
or
(2.2) |G| = 2, 2r2 = 2r1 for all r1 € R1 (in addition 4 = 0 in R1), and R2 = {0} or R2 is a ring of char(R2) = 3.
Proof. If G is trivial, there is nothing to prove because of the isomorphism R[G] = R, so let us assume henceforth that G is non-trivial.
"Necessity." As already observed in Proposition 2 alluded to above, if G = {1}, then the ring R must be invo-clean but not properly weakly invo-clean, i.e., it does not contain Z5 as a (proper) direct factor. Thus R[G] has to be invo-clean too, as char(R[G]) = char(R). We, therefore, appeal to Proposition 1 getting the listed above two items, as desired.
"Sufficiency." As in the previous direction, Proposition 1 is in use to infer that R[G] is invo-clean and hence weakly invo-clean, as wanted. □
In closing, we state one more intriguing problem.
Problem 1. Find a suitable criterion only in terms of the commutative unital ring R and the abelian group G when the group ring R[G] is feebly invo-clean as defined in [3].
In that direction, similarly to Lemma 3, the question of whether or not any (commutative) feebly invo-clean ring R which is possibly not weakly invo-clean possesses the decomposition R = K x P, where K is a weakly invo-clean ring and P is a ring whose elements satisfy the equation x5 = x such that P = Z5, is of some interest.
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