RESEARCH ARTICLE

Serial piecewise domains

Nadiya Gubareni^a, Marina Khibina<sup>b

a</sup> Institute of Econometrics & Computer Science, Technical University of Częestochowa, 42–200 Częestochowa, Poland
^b Institute of Engineering Thermophysics, NAS, Kiev, Ukraine

Abstract: A ring $A$ is called a piecewise domain with respect to the complete set of idempotents $\{e_1, e_2,\ldots, e_m\}$ if every nonzero homomorphism $e_iA \rightarrow e_jA$ is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivalent. We prove that a serial right Noetherian ring is a piecewise domain if and only if it is right hereditary. And we prove that a serial ring with right Noetherian diagonal is a piecewise domain if and only if it is semihereditary.

Keywords: piecewise domain, hereditary ring, semihereditary ring, serial ring, Noetherian diagonal, prime radical, prime quiver.

MSC: 16P40, 16G10

Language: English

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