# Notes on reduced Rickart rings, I

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## Abstract

A reduced Rickart ring is considered as a reduced ring *R* with an additional operation which associates to every element \(a \in R\) the single idempotent *e* such that the ideal *eR* is the right annihilator of *a*. We discuss some elementary properties of this operation, prove that a ring is reduced and Rickart if and only if it is isomorphic to an associate ring in the sense of I. Sussman (a certain subdirect product of domains with “enough” idempotents), and present several equational axiom systems for reduced Rickart rings.

## Keywords

Equational axioms Focal operation Reduced ring Rickart ring Subdirect product Sussman ring## Mathematics Subject Classification

16W99 16S60 16R10## Notes

### Acknowledgements

The authors are grateful to the anonymous referee for his/her remarks concerning the presentation and for suggestions that have resulted in Example 2.1 and Proposition 2.3. The first author was partially supported by Latvian Council of Science, Grant No. 271/2012.

### Compliance with ethical standards

#### Conflict of interest

The authors declare that they have no conflict of interest. The first author has not any financial relationship with the supporting organization.

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