“The Human Was Created Out of Haste.” On Prophecy and the Problem of Human Nature in the Qur’an

In this article it is argued that the Qur’an’s doctrine of divine mercy is best understood in light of its pessimistic anthropology, an aspect of the text that is often underappreciated. The so-called “primordial covenant” verse (Q 7:172) of the Qur’an holds humans responsible for submission to God. The Qur’anic language on “signs” in the natural world suggests that humans should recognize God (and be grateful to Him) by reflection on nature alone. Yet, according to the Qur’an they do not. The Qur’an refers frequently to humans as “ungrateful” and “hasty”. It also makes divine punishments a regular element of human history, suggesting that rebellion is endemic to human nature. It is, I argue, precisely the rebelliousness of humans that makes God’s initiative in sending prophets merciful. The ministry of prophets in the Qur’an is an unmerited manifestation of divine compassion for a sinful humanity.

Inverse complements and strongly unit regular elements

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

Clear rings and clear elements

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

TRIPLET INVARIANCE AND PARALLEL SUMS

Let R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$ . Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$ , then $I(a)+I(b)=I({\cal P}(a, b))$ . Conversely, if $I(a)+I(b)=I(c)$ for some $c\in R$ , then $\mathrm {E}[c]a(a+b)^{-}b$ is invariant for all $(a+b)^{-}\in I(a+b)$ , where $\mathrm {E}[c]$ is the smallest idempotent in C satisfying $c=\mathrm {E}[c]c$ . This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.

A note on r-precious ring

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

Positive Entropy Using Hecke Operators at a Single Place

We prove the following statement: let $X=\textrm{SL}_n({{\mathbb{Z}}})\backslash \textrm{SL}_n({{\mathbb{R}}})$ and consider the standard action of the diagonal group $A<\textrm{SL}_n({{\mathbb{R}}})$ on it. Let $\mu $ be an $A$-invariant probability measure on $X$, which is a limit $$\begin{equation*} \mu=\lambda\lim_i|\phi_i|^2dx, \end{equation*}$$where $\phi _i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$ and $\lambda>0$ is some positive constant. Then any regular element $a\in A$ acts on $\mu $ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over ${{\mathbb{Q}}}$ and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss [2].

Matrix factorizations for self-orthogonal categories of modules

For a commutative ring [Formula: see text] and self-orthogonal subcategory [Formula: see text] of [Formula: see text], we consider matrix factorizations whose modules belong to [Formula: see text]. Let [Formula: see text] be a regular element. If [Formula: see text] is [Formula: see text]-regular for every [Formula: see text], we show there is a natural embedding of the homotopy category of [Formula: see text]-factorizations of [Formula: see text] into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if [Formula: see text] is the category of projective or flat-cotorsion [Formula: see text]-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when [Formula: see text] is the category of injective [Formula: see text]-modules.

Partial Addition and Ternary Product Based -So-Semirings-2

Here we are introducing thenotions i-system, idempotent, centre of a ternary  -SO semiring, Nilpotent are introduced and it is proved that some equivalent conditions. Further it is also proved that (i) if C be a ternary  - SO semiring, m is a “strongly regular element”, then ∃𝝑, 𝝁∈Г also n∈C ∋m = m𝝑n𝝁m,n = n𝝁m𝝑n (ii) If “I be an Ideal of A strongly regular ternary  - SOsemiring R then I is strongly regular and any ideal J of I is an ideal of R” and many more properties were proved. Mathematical subject classification: 16Y60.

Some splitting theorems for local cohomology modules with respect to a pair of ideals

Suppose that [Formula: see text] is a commutative ring with identity, [Formula: see text], [Formula: see text] are ideals of [Formula: see text], and let [Formula: see text] be a finitely generated [Formula: see text]-module. Let [Formula: see text] be the [Formula: see text]th local cohomology functor with respect to [Formula: see text]. In this paper, for fixed integers [Formula: see text] and [Formula: see text], we study the existence of the following isomorphisms of local cohomology modules: (i) [Formula: see text]; (ii) [Formula: see text]; and, (iii) [Formula: see text] for some filter regular element [Formula: see text] on [Formula: see text]. Moreover, we provide some applications of the above isomorphisms.

Cohen Macaulay Bipartite Graphs and Regular Element on the Powers of Bipartite Edge Ideals

In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals I ( G ) , we also discuss methods to recognize regular elements on I ( G ) s for all s ≥ 1 in terms of the combinatorics of the graph G.