Narrow Results

For each commutative ring $R$ we associate a simple graph ${\mathrm{\Gamma }}_1^{\ast }(R)$. We investigate the properties of an Artinian ring $R$ when ${\mathrm{\Gamma }}_1^{\ast }(R)$ is a star.

Let K be an algebraic extension of a field k, and let G be a direct product of finitely many cyclic groups. We describe a method of construction of maximal ideals m satisfying kG/m ≅ K where kG is the group algebra of G over k. Moreover, we give examples by using Computer Algebra Systems.

Let R be a finite commutative ring with identity. The co-maximal graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also, Γ₂(R) is the subgraph of Γ(R) induced by non-unit elements and ${{\Gamma }^{\prime }}_2\left(R\right)={\Gamma }_2\left(R\right)\backslash J\left(R\right)$ where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which ${{\Gamma }^{\prime }}_2\left(R\right)$, Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.

Minimal and maximal right ideals in the nearring R[x] are identified where R is a commutative ring with identity. Particular attention is given to the case where the maximal right ideals contain maximal two-sided ideals and where R is a finite field.

A (commutative integral) domain R is said to be valuative if, for each nonzero element u in the quotient field of R, at least one of R ⊆ R[u] and R ⊆ R[u^-1] has no proper intermediate rings. Such domains are closely related to valuation domains. If R is a valuative domain, then R has at most three maximal ideals, and at most two if R is not integrally closed. Also, if R is valuative, the set of nonmaximal prime ideals of R is linearly ordered, at most one maximal ideal of R does not contain each nonmaximal prime of R, and R_P is a valuation domain for each prime P except for at most one maximal ideal. Any integrally closed valuative domain is a Bézout domain. Valuation domains are characterized as the quasilocal integrally closed valuative domains. Each one-dimensional Prüfer domain with at most three maximal ideals is valuative.

The first aim of this article is to study maximal ideals of a preadditive category . Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If is additive and semilocal, any skeleton of is a Krull monoid and we are able to characterize the essential valuations of and provide some natural divisor homomorphisms and divisor theories of .

Some properties of (left) k-ideals and r-ideals of a semiring are considered by the help of the congruence class semiring. It is proved that a proper k-ideal of a semiring with an identity is prime if it is a maximal left k-ideal. An equivalent condition for a proper r-ideal of a semiring being a maximal (left) r-ideal is established. It is shown that (left) r-ideals and (left) k-ideals coincide for an additively idempotent semiring, though the former is a special kind of the latter in general. It is proved that a proper k-ideal of an incline with an identity is a maximal k-ideal if and only if the corresponding congruence class semiring is the Boolean semiring.

In this paper, we describe the maximal ideals of the endomorphism ring of an injective module.

In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

Given any minimal ring extension $k\subset L$ of finite fields, several families of examples are constructed of a finite local (commutative unital) ring $A$ which is not a field, with a (necessarily finite) inert (minimal ring) extension $A\subset B$ (so that $B$ is a separable $A$-algebra), such that $A\subset B$ is not a Galois extension and the residue field of $A$ (respectively, $B$) is $k$ (respectively, $L$). These results refute an assertion of G. Ganske and McDonald stating that if $R\subseteq S$ are finite local rings such that $S$ is a separable $R$-algebra, then $R\subseteq S$ is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let $(A,M)$ be a finite special principal ideal ring (SPIR), but not a field, such that $M$ has index of nilpotency $\alpha$ ($\ge 2$). Impose the uniform distribution on the (finite) set of ($A$-algebra) isomorphism classes of the minimal ring extensions of $A$. If $2\in M$ (for instance, if $A\cong \mathbb{Z}/2^{\alpha }\mathbb{Z}$), the probability that a random isomorphism class consists of ramified extensions of $A$ is at least $2/3$; if $2\notin M$ (for instance, if $A\cong \mathbb{Z}/p^{\alpha }\mathbb{Z}$ for some odd prime $p$), the corresponding probability is at least $3/4$. Additional applications, examples and historical remarks are given.

We prove that in any $J$-Noetherian Bezout domain which is not of stable range 1, there exists a nonunit adequate element (element of almost stable range 1).

For a subset $T$ of a ring $R$ we denote by ${ℐ}(T)$ the ideal of $R$ generated by $T$. Given a higher commutator $L$ of $R$, if $R={ℐ}(L)$ then $R=L+L^2$? The question is motivated by the result that a ring $R$ is equal to its subring generated by $[R,R]$ if $R$ is either a noncommutative simple ring (by Herstein) or a unital ring with $1\in [R,R]$ (by Eroǧlu). In this note, we study the question for the rings $R$ satisfying the property that every proper ideal of $R$ is contained in a maximal ideal (in particular, if $R$ is finitely generated as an ideal).

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

Motivated by topological approaches to Euclid and Dirichlet's theorems on infinitude of primes, we introduce and study -coprime topologies on a commutative ring R with an identity and without zero divisors. For infinite semiprimitive commutative domain R of finite character (i.e. every nonzero element of R is contained in at most finitely many maximal ideals of R), we characterize its subsets A for which the Dirichlet condition, requiring the existence of infinitely many pairwise nonassociated elements from A in every open set in the invertible topology, is satisfied.

A dually normal almost distributive lattice is characterized topologically in terms of its maximal ideals and prime ideals. Some necessary and sufficient conditions for the space of maximal ideals to be dually normal are obtained.

A regular double Stone algebra with nonvoid core is called core regular double Stone algebra (CRDSA) [6] and this particular core element affects the behavior of the algebra in certain aspects especially in characterization of maximal and prime ideals. In this paper, an elegant characterization for maximal and prime ideals of a CRDSA is established.