The group of units in a compact ring

Cohen, Jo-Ann; Koh, Kwangil

doi:10.1016/0022-4049(88)90028-x

Cited by 9 publications

(5 citation statements)

References 6 publications

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“…Choose a,b € R such that ab = 1 and choose g € G. Then a(bn) = n and therefore agbn = gn, so agb = g, because n is a unit. So, Lemma 1.1 implies that G is Abelian and therefore R is commutative by [1,Theorem 3.2]. This, together with the existence of ei, is a contradiction with the indecomposability of R. Therefore, we can conclude that R is indeed a local ring.…”

Section: The Properties Of N-insertive Rings Theorem 2 1 Let N Bementioning

confidence: 93%

“…On the other hand, if G is commutative, then R is commutative by a corollary of [1,Theorem 3.2]. This implies that R, and then of course also G, is 1-insertive.…”

Section: Lemma 1 1 G Is L-insertive If and Only If G Is Commutativementioning

confidence: 98%

“…By the x=i previous lemma we conclude that all elements in J are also in the centraliser of G, thus 1 + J is a commutative group, so J is commutative as well. Thus G is an Abelian group and therefore R is a commutative ring by the corollary of [1,Theorem 3.2]. D The next example shows that this theorem does not hold if n is not a unit.…”

mentioning

confidence: 95%

“…For example, it was shown in [1] that a finite ring is commutative if and only if its group of units is commutative. The notion of commutativity can be generalised to the notion of n-insertiveness, as shown below.…”

Section: Introductionmentioning

confidence: 99%

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The n-insertive subgroups of units

Dolžn

2007

Bull. Austral. Math. Soc.

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Let R be a finite ring. Let us denote its group of units by G = G[R) and its Jacobson radical by J = J(R). Let n be an arbitrary integer. We prove that R is an n-insertive ring if and only if G is an n-insertive group and show that every n-insertive finite ring is a direct sum of local rings. We prove that if n is a unit, then the local ring R is n-insertive if and only if its Jacobson group 1 + J is n-insertive and find an example to show that this is not true if n is a non-unit.

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Section: The Properties Of N-insertive Rings Theorem 2 1 Let N Bementioning

confidence: 93%

“…On the other hand, if G is commutative, then R is commutative by a corollary of [1,Theorem 3.2]. This implies that R, and then of course also G, is 1-insertive.…”

Section: Lemma 1 1 G Is L-insertive If and Only If G Is Commutativementioning

confidence: 98%

mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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The n-insertive subgroups of units

Dolžn

2007

Bull. Austral. Math. Soc.

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“…The standard example of a noncommutative ring R with commutative R * is the free associative algebra K x, y in noncommuting variables x, y over a field K. For other examples we refer to [17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

A note on commutators in the group of infinite triangular matrices over a ring

Bier

Hołubowski

2015

Linear and Multilinear Algebra

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We investigate the commutators of elements of the group UT(∞, R) of infinite unitriangular matrices over an associative ring R with 1 and a commutative group R * of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence we give a complete characterization of the lower central series of the group UT(∞, R) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring R, we show that the derived subgroup of T(∞, R) coincides with the group UT(∞, R). The obtained results generalize the results obtained for triangular groups over a field.

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