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Our aim in this paper is to explore semisubtractive ideals of semirings. We prove that they form a complete modular lattice. We introduce Golan closures and prove some of their basic properties. We explore the relations between $Q$-ideals and semisubtractive ideals of semirings, and also study them in $s$-local semirings. We introduce two subclasses of semisubtractive ideals: $s$-strongly irreducible and $s$-irreducible, and provide various representation theorems. By endowing a topology on the set of semisubtractive ideals, we prove that the space is $T_0$, sober, connected, and quasi-compact. We also briefly study continuous maps between semisubtractive spaces. We construct $s$-congruences and prove a bijection between these congruences and semisubtractive ideals., Comment: 13 pages

The purpose of this note is to study some algebraic properties of irreducible ideals of monoids. We establish relations between irreducible, prime, and semiprime ideals. We explore some properties of irreducible ideals in local, Noetherian, and Laskerian monoids., Comment: 7 pages

In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.

We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, encompassing lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space, and we further explore continuous maps between lower spaces., Comment: arXiv admin note: text overlap with arXiv:2308.06395

We prove that the set of proper ideals of a monoid endowed with coarse lower topology is a spectral space.

The aim of this paper is to investigate further properties of $z$-elements in multiplicative lattices. We utilize $z$-closure operators to extend several properties of $z$-ideals to $z$-elements and introduce various distinguished subclasses of $z$-elements, such as $z$-prime, $z$-semiprime, $z$-primary, $z$-irreducible, and $z$-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where $z$-elements are closed under finite products and a representation of $z$-elements in terms of $z$-irreducible elements in $z$-Noetherian multiplicative lattices., Comment: 15 pages

The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices introduced in D. D. Anderson, C. Jayaram, and P. A. Phiri, Baer lattices, \textit{Acta Sci. Math. (Szeged)}, 59 (1994), 61--74. Additionally, we present results concerning these elements that have not been addressed in the study of $d$-ideals of rings. Furthermore, we introduce Baer closures and explore Baer maximal, prime, semiprime, and meet-irreducible elements., Comment: 12 apges

As a natural extension of the ongoing development of a theory of ideals in commutative quantales with an identity element, this article aims to study into the analysis of certain topological properties exhibited by distinguished classes of ideals. These ideals are equipped with quantale topologies. The primary objectives encompass characterizing quantale spaces exhibiting sobriety, examining several conditions pertaining to quasi-compactness, and demonstrating that quantale spaces comprised of proper ideals adhere to the spectral properties as defined by Hochster. We introduce the notion of a strongly disconnected spaces and show that for a quantale with zero Jacobson radical, strongly disconnected quantale spaces containing all maximal ideals of the quantale imply existence of non-trivial idempotent elements in the quantale. Additionally, a sufficient criterion for establishing the connectedness of a quantale space is presented. Finally, we discuss on continuous maps between quantale spaces.

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of several key properties related to ideals in quantale, including prime, semiprime, radical, primary, irreducible, and strongly irreducible ideals. Furthermore, we investigate the primary decomposition problem for quantale ideals. In conclusion, we present a set of future directions for further exploration, serving as a natural continuation of this article.

The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to strongly irreducible ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are $T_0$, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize arithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of maximal and prime ideals to be dense in the corresponding terminal spaces.

We study the topology of a class of proper submodules and some of its distinguished subclasses and call them structure spaces. We give several criteria for the quasi-compactness of these structure spaces. We study $T_0$ and $T_1$ separation properties and characterize structure spaces in which nonempty irreducible closed subsets have unique generic points. We provide a sufficient condition for the connectedness of structure spaces. We prove that the structure spaces of proper submodules are spectral, and moreover, we characterize the spectral structure spaces of Noetherian modules. Finally, we discuss continuous functions between these spaces.

The aim of this paper is to study some distinguished classes of $k$-ideals of semirings, which include $k$-prime, $k$-semiprime, $k$-radical, $k$-irreducible, and $k$-strongly irreducible ideals. We discuss some of the properties of $k$-ideals and their products, intersections, and ideal quotients under semiring homomorphisms.

Using the closure operator that defines a subtractive ideal of a semiring $S$, in this note we introduce a topology on the set of all ideals of $S$ induced by that operator. We show that the corresponding subtractive space is $T_0$ and every nonempty irreducible closed set has a unique generic point, whereas the restricted subspace of subtractive ideals is $T_1$. Using a semiring homomorphism, we obtain a continuous map between the corresponding subtractive spaces.

Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. As remarked in the prequel of the present paper (`Duality in Non-Abelian Algebra IV'), there is a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. It is based on the axiomatic context of a `noetherian form' introduced in the same paper, which is a generalization of an abelian category that includes among its examples semi-abelian and Grandis exact categories, as well as many others. The aim of the present article is to expose the proofs of homological diagram lemmas in this general context.

The aim of this paper is to study the primitive ideals of Novikov algebras. In terms of modular maximal right ideals, a characterization of the primitive ideals of a Novikov algebra has been obtained. We prove a Chevalley-Jacobson density-type theorem for primitive Novikov algebras. We obtain some equivalences between prime, simple, and primitive Novikov algebras. We describe a subalgebra of a Novikov algebra as a Novikov algebra of endomorphisms., Comment: There are some flaws and mistakes

The aim of this paper is to study the topological properties of some classes of subsemimodules endowed with a subbasis closed-set topology. We show that such spaces are $T_0$. When the semimodule is finitely generated, those spaces are compact as well. We characterize subsemimodule spaces for which every nonempty irreducible closed set has a unique generic point. We give a sufficient condition for a connected subsemimodule space, and using the notion of strongly disconnectedness, we determine compact disconnected subsemimodule spaces. Finally, we discuss continuous maps between subsemimodule spaces.

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the corresponding Zariski space. We prove that a $k$-space is $T_{\scriptscriptstyle 0}$, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such $k$-spaces. We conclude with a question about construction of a sheaf of $k$-spaces similar to affine schemes.

The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We show arithmetic ring has infinite Krull dimension and its associated prime ideal set is nonempty., Comment: The paper contains some incorrect results

Using a new definition of a prime ideal of a skew brace A, on set Spec A of prime ideals of A we endow a spectral topology (in the sense of Grothendieck). We characterize irreducible closed subsets of Spec A and prove every irreducible closed subset of the space has a unique generic point. We give a sufficient condition for the space to be Noetherian. We study continuous maps between such spaces, and finally, we prove that Spec(Idl A) is a spectral space, where Idl A is the set of all ideals of A.

Can we do a topological study of various classes of normal subgroups endowed with a hull-kernel-type topology? In this paper, we have provided an answer to this question. We have introduced as well a new class of normal subgroups called primitive subgroups. Separation axioms, compactness, connectedness, and continuities of these spaces have been studied. We have concluded with the question of determining spectral spaces among them.

Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a Lie algebra. We give several Lie-module-related characterizations of primitive Lie algebras and prove a representation theorem of a Lie algebra in terms of primitive Lie algebras. We prove a density theorem. Finally, we highlight on the importance of nuclear ideals of prime and semiprime Lie algebras in determining them., Comment: There are some flaws and mistakes

We introduce primitive hyperideals of a hyperring R and show relations with R itself, and with maximal and prime hyperideals of R. We endow a Jacobson topology on the set of primitive hyperideals of R and study topological properties of the corresponding hyperstructure space.

The aim of this paper is to study Iseki spaces of distinguished classes of ideals of a semiring endowed with a topology. We show that every Is\'{e}ki space is quasi-compact whenever the semiring is Noetherian. We characterize Is\'{e}ki spaces for which every non-empty irreducible closed subset has a unique generic point. Furthermore, we provide a sufficient condition for the connectedness of Is\'{e}ki spaces and show that the strongly connectedness of an Is\'{e}ki space implies the existence of non-trivial idempotent elements of semirings.

The aim of this paper is to describe a population model with transition. We analyze the spectral properties of the transition matrix considering both irreducible and reducible structures. We give physical interpretations of these properties to population dynamics.

The purpose of this note is to introduce primitive ideals of noncommutative semigroups and study some topological aspects of the corresponding structure spaces.

We consider the set of all the ideals of a ring, endowed with the coarse lower topology. The aim of this paper is to study the topological properties of distinguished subspaces of this space and detect the spectrality of some of them.

Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we intend to show that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space., Comment: 4 pages

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of non-trivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper we establish such a generalization of the "salamander lemma" due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Marki and W. Tholen and exact categories in the sense of M. Grandis., Comment: 11 pages

The notion of L-homologies (of double complexes) as proposed in this paper extends the notion of classical horizontal and vertical homologies, along with two other new homologies introduced in the homological diagram lemma called salamander lemma. We enumerate all L-homologies associated with an object of a double complex and provide new examples of exact sequences. We describe a classification problem of these exact sequences. We study two poset structures on these L-homologies; one of them determines the trivialities of horizontal and vertical homologies of an object in terms of other L-homologies of that object, whereas the second structure shows the significance of the two homologies introduced in salamander lemma. Finally, we prove the existence of a faithful amnestic Grothendieck fibration from the category of L-homologies to a category consisting of objects and morphisms of a given double complex., Comment: 10 pages