Irreducible Ring Research Articles

Strongly irreducible ring and its S spectrum

Let Κ be a proper ideal of a commutative ring S. Then Κ is a strongly irreducible (SI) ideal if for any two ideals C and D of S, C∩D⊆ Κ implies C ⊆ Κ or D ⊆ Κ. We say a ring is strongly irreducible (SI) if all its ideals are strongly irreducible. In this paper some properties of such ring are given. The relations between SI rings and some types of rings are also studied. Furthermore, for an SI ring S we study the S spectrum of S and the S.spec(S) topology.

Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a , b ∈ R \\ { 0 } . The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I ⊆ RaR of R or aJ = 0 for some nonzero ideal J ⊆ RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I ⊆ RaR and J ⊆ RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.

Characterizing certain semidualizing complexes via their Betti and Bass numbers

Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let C be a semidualizing complex for an analytically irreducible local ring R and set and We show that C is quasi-isomorphic to a shift of R if and only if the nth Betti number of C is one. Also, we show that C is a dualizing complex for R if and only if the dth Bass number of C is one.

R+ˆ is surprisingly an integral domain

It is shown that if ( R , m , k ) is a complete local domain with char k = p > 0 and R + is its integral closure in an algebraic closure of the quotient field, then both the m -adic and p -adic completions of R + are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R -modules (even balanced in the m -adic case), will have dimension larger than the dimension of R unless dim ⁡ R ≤ 1 .

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.

Irreducibility and Reduction Number of Ideals in a One Dimensional Analytically Irreducible Ring

Let R be a one dimensional analytically irreducible ring, and let I be an integral ideal of R. We study the irreducibility and the reduction number of I in relation with the corresponding semigroup ideal v(I) in v(R), where v(R) is the semigroup of values of R. It turns out that, if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We show also that the reduction number of I in R can assume any positive value less than the multiplicity of R and can be different from the reduction number of v(I).

The analogue of Izumi's Theorem for Abhyankar valuations

A well-known theorem of Izumi, strengthened by Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring ( R , m ) are linearly comparable to each other. This is equivalent to saying that any divisorial valuation ν centered in R is linearly comparable to the m-adic order. In the present paper, we generalize this theorem to the case of Abhyankar valuations ν with archimedian value semigroup Φ. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. In other words, saying that R is analytically irreducible, ν is Abhyankar and Φ is archimedian is equivalent to linear equivalence of topologies plus another condition called weak noetherianity of the graded algebra gr ν R . We give some applications of Izumi's Theorem and of Lemma 2.8, which is a crucial step in our proof of the main theorem. We show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the m-adic topology and the topology defined by the symbolic powers of an arbitrary ideal.

Irreducibility of ideals in a one-dimensional analytically irreducible ring

Irreducibility of ideals in a one-dimensional analytically irreducible ring

A KUROSH-AMITSUR LEFT JACOBSON RADICAL FOR RIGHT NEAR-RINGS

Let R be a right near-ring. An R-group of type-5/2 which is a natural generalization of an irreducible (ring) module is introduced in near-rings. An R-group of type-5/2 is an R-group of type-2 and an R-group of type-3 is an R-group of type-5/2. Using it <TEX>$J_{5/2}$</TEX>, the Jacobson radical of type-5/2, is introduced in near-rings and it is observed that <TEX>$J_2(R){\subseteq}J_{5/2}(R){\subseteq}J_3(R)$</TEX>. It is shown that <TEX>$J_{5/2}$</TEX> is an ideal-hereditary Kurosh-Amitsur radical (KA-radical) in the class of all zero-symmetric near-rings. But <TEX>$J_{5/2}$</TEX> is not a KA-radical in the class of all near-rings. By introducing an R-group of type-(5/2)(0) it is shown that <TEX>$J_{(5/2)(0)}$</TEX>, the corresponding Jacobson radical of type-(5/2)(0), is a KA-radical in the class of all near-rings which extends the radical <TEX>$J_{5/2}$</TEX> of zero-symmetric near-rings to the class of all near-rings.

A tight closure analogue of analytic spread

An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a “Nakayama lemma for tight closure”. Then, after strengthening A. Vraciu's theory of *-independence and the special part of tight closure, it is shown that all minimal *-reductions of an ideal in an analytically irreducible excellent local ring of positive characteristic have the same minimal number of generators. This number is called the *-spread of the ideal, by analogy with the notion of analytic spread.

On the finiteness of Bass numbers of local cohomology modules

In this note we show that, if I I is an ideal of dimension 1 of an analytically irreducible local ring, then the Bass numbers of local cohomology modules with support in V ( I ) V(I) are finite.

Modeling Amorphization of Tetrahedral Structures Using Local Approaches

AbstractMany crystalline ceramics can be topologically disordered (amorphized) by disordering radiation events involving high-energy collision cascades or (in some cases) successive single-atom displacements. We are interested in both the potential for disorder and the possible aperiodic structures adopted following the disordering event. The potential for disordering is related to connectivity, and among those structures of interest are tetrahedral networks (such as SiO2, SiC and Si3N4) comprising corner-shared tetrahedral units whose connectivities are easily evaluated. In order to study the response of these networks to radiation, we have chosen to model their assembly according to the (simple) local rules that each corner obeys in connecting to another tetrahedron; in this way we easily erect large computer models of any crystalline polymorphic form. Amorphous structures can be similarly grown by application of altered rules. We have adopted a simple model of irradiation in which all bonds in the neighborhood of a designated tetrahedron are destroyed, and we reform the bonds in this region according to a set of (possibly different) local rules appropriate to the environmental conditions. When a tetrahedron approaches the boundary of this neighborhood, it undergoes an optimization step in which a spring is inserted between two corners of compatible tetrahedra when they are within a certain distance of one another; component forces are then applied that act to minimize the distance between these corners and minimize the deviation from the rules. The resulting structure is then analyzed for the complete adjacency matrix, irreducible ring statistics, and bond angle distributions.

Polynomial rings over Jacobson-Hilbert rings

All rings considered are commutative with unit. A ring R is SISI (in Vamos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. SISI rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a SISI ring R is again SISI. In this paper we show this is not the case.

On the additive groups of subdirectly irreducible rings

In this paper we study the additive group structure of subdirectly irreducible rings and their hearts. We give and example of a torsion-free, non-reduced abelian group which is not the underlying additive group of any associative subdirectly irreducible ring. It is a counterexample to a theorem in Feigelstock's book “Additive Groups of Rings.”

A note on subdirectly irreducible rings

Let R be a commutative subdirectly irreducible ring, with minimal ideal M. It is shown that either R is a field, or M2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.

The additive groups of subdirectly irreducible rings

An abelian group G is said to be subdirectly irreducible if there exists a subdirectly irreducible ring R with additive group G. If G is subdirectly irreducible, and if every ring R with additive group G, and R2 ≠ 0, is subdirectly irreducible, then G is said to be strongly subdirectly irreducible. The torsion, and torsion free, subdirectly irreducible and strongly subdirectly irreducible groups are classified completely. Results are also obtained concerning mixed subdirectly irreducible and strongly subdirectly irreducible groups.

Equal-Time Commutation Relations for Heisenberg Fields in the Lee Model

>~ 4 > has been successfully applied to problems in both high energy physics5> and the area of many-body problems, particularly in superconductivity 6> and ferromagnetism. 7> In this paper we apply this method to the solvable Dirac­ Lee model which, unlike the pair model considered in Ref. 3), has mass and coupling constant renormalizations. The equal-time commutation relations among Heisenberg fields are derived and not assumed. Further we show that the wave­ function renormalization constant is determined from microcausality. This result sheds light on the question why in Ref. 3) the existence of the~ bound state was found to be closely connected with microcausality. The self-consistent method exploits the duality between basic fields and observed particles concretely exhibited in the presence of interactions. The basic Heisenberg fields obey nonlinear field equations which reflect the laws of nature, while the physical particles are the ones that appear in observations. These physical fields obey free field equations and define the physical Fock space. The field equations for the Heisenberg fields are to be regarded as operator equations in this physical Fock space. The Heisenberg field is related to the physical field by means of a mapping known as the dynamical map.4> The set ?f physical fields is taken to form an irreducible operator ring. Therefore, any operator of the Fock space can be written as a sum of normal products of physical flelds. The ability of this method to predict new particles rests on this postulate. The asymptotic condition may tell us that the physical fields are not complete. The completion of this set of fields then requires the existence of other particles at the level of observation and should be incorporated. 8>·8> The success of this method therefore rests on the proper choice of the set of physical fields.

Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents

It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.

A note on subdirectly irreducible rings

It is proved that every subdirectly irreducible ring can be obtained as a proper homomorphic image of another subdirectly irreducible ring. An example of a subdirectly irreducible ring R with heart H is given for which i) R has a non-subdirectly irreducible homomorphic image, ii) R/H is an integral domain, and iii) the commutator ideal C(R) of R coincides with R.

Primitive rings with involution whose symmetric elements satisfy a generalized polynomial identity

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n: