Sums of units in Baer and exchange rings

Document Type : Research Paper

Authors

1 Faculty of Engineering, Shohadaye Hoveizeh Campus of Technology, Shahid Chamran University of Ahvaz, Khuzestan, Iran.

2 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-3619995161, Shahrood, Iran

Abstract

In this paper, we prove that every element in an exchange ring $R$ with artinian primitive factors is $n$-tuplet-good iff $1_R$ is $n$-tuplet-good. Also, we show that for such rings the full matrix ring $M_n(R)$ (for $n\geq 2$) is $n$-tuplet-good. In [7], Fisher and Snider proved that every element of a strongly $\pi$-regular ring $R$ with $\frac{1}{2}\in R$ is 2-good. We prove the same result under new condition and show that these rings are twin-good. We also consider the conditions under which endomorphism ring of a finitely generated projective module $M$ over unit regular ring $L$ is 2-tuplet-good. The main result of [14] states that regular self-injective rings are $n$-tuplet-good if such rings has no factor ring isomorphic to a field $D$ with $|D|<n+2$. We generalized this result to regular Baer rings proving that every regular Baer ring $R$ that has no factor ring isomorphic to a field of order less than $n+2$, is $n$-tuplet-good.

Keywords

References

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Algebraic Structures and Their Applications
Volume 10, Issue 1
February 2023
Pages 47-55

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