The singular ideal and the socle of incidence rings

Abstract

Let $R$ be a ring with identity and $I(X,R)$ be the incidence ring of a locally finite partially ordered set $X$ over $R.$ In this paper, we compute the socle and the singular ideal of the incidence ring for some $X$ in terms of the socle of $R$ and the singular ideal of $R$, respectively.

Keywords

• incidence algebras
• socle
• singular ideal

References

1. [1] F. Al-Thukair, S. Singh and I. Zaguia, Maximal ring of quotients of an incidence algebra, Arch. Math. 80, 358–362, 2003.
2. [2] S. Esin, M. Kanuni and A. Koç, Characterization of some ring properties in incidence algebras, Comm. Algebra, 39 (10), 3836–3848, 2011.
3. [3] M. Kanuni, Dense ideals and maximal quotient rings of incidence algebras, Comm. Algebra, 31 (11), 5287–5304, 2003.
4. [4] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, New York-Berlin, Springer-Verlag, 1999.
5. [5] E. Spiegel, Essential ideals of incidence algebras, J. Austral. Math. Soc. (Series A), 68, 252–260, 2000.
6. [6] E. Spiegel and C.J. O’Donnell, Incidence Algebras, Monographs and Textbooks in Pure Appl. Math. 206, New York, Marcel Dekker, 1997.