D.C.C. rings with a cyclic group of units

Eldridge, Klaus E.; Fischer, Irwin

doi:10.1215/s0012-7094-67-03428-x

Cited by 18 publications

(10 citation statements)

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“…Gilmer [3] determines all finite commutative rings whose group of units is cyclic. Eldridge and Fischer [2] extend these results to artinian rings, and, in [1], Eldridge shows to what extent the structure of an artinian ring is determined by knowing that it has either a solvable, simple, nilpotent, supersolvable, torsion, or finitely generated group of units.…”

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confidence: 85%

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On the Group of Units of a Ring

Ditor¹

1971

The American Mathematical Monthly

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confidence: 85%

“…Hence the subring [G] generated by G is a finite-dimensional algebra over GF (2). Hence the subring [G] generated by G is a finite-dimensional algebra over GF (2).…”

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confidence: 99%

On the Group of Units of a Ring

Ditor¹

1971

The American Mathematical Monthly

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“…3* Cyclic groups of units* Gilmer [4] has characterized all finite commutative rings with a cyclic group of units and Eldridge and Fischer [3] have extended these results to artinian rings. In order to cover the semiperfect case we need the following negative result.…”

Section: Let R Be a Semiperfect Ring With Jβ* Abelian If R* Is Finitmentioning

confidence: 99%

Semiperfect rings with abelian group of units

Nicholson¹

1973

Pacific J. Math.

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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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D.C.C. rings with a cyclic group of units

Cited by 18 publications

References 0 publications

On the Group of Units of a Ring

On the Group of Units of a Ring

Semiperfect rings with abelian group of units

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

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