Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by −−→ Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J , there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), h...

The susceptibility of ring networks to disconnection as a result of one or two node/link failures has led to a number of proposals to increase the robustness of such networks. One class of proposals, discussed primarily in the networking and communication communities, but also advocated for use in parallel and distributed systems, involves the provision of a second ring to improve system throug...

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G = GantGaff ∼= (Gant ×Gaff )/Gaff ∩Gant, where Gant is the maximal anti-affine subgroup of G, and Gaff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M = GantMaff ∼= Gant∗Gaff∩GantMaff , whereMaff i...

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Necessary and sufficient conditions for an element in an incline to be regular are obtained. It is proved that every regular incline is a distributive lattice. The existence of the Moore-Penrose inverse of an element in an incline with involution is discussed. Characterizations of the set of ...

We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). 2000 Mathematics Subject Classification. Primary 16E50, 19B10. An ...

Gennady Lyubeznik conjectured that if R is a regular ring and a is an ideal of R, then the local cohomology modules H i a(R) have only finitely many associated prime ideals, [Ly1, Remark 3.7 (iii)]. While this conjecture remains open in this generality, several results are now available: if the regular ring R contains a field of prime characteristic p > 0, Huneke and Sharp showed in [HS] that t...

The Kerr black holes possess a photon region with prograde and retrograde orbits radii, respectively, $M\leq r_p^-\leq 3M$ $3M\leq r_p^+\leq 4M$, thereby always cast closed ring or shadow silhouette for $a\leq M$. For $a>M$, it is no-horizon spacetime (naked singularity) wherein spiral into the central singularity, produce an arc-like dark spot at center. We compare holes' structure those produ...