Projective geometry over rings with stable range condition

A generalization of the fundamental theorem of projective geometry is established for non-injective mappings between projective geometries defined over rings which satisfy a stable range condition. A geometrical characterization of the stable range condition is also given.     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

Group rings over Dedekind rings

Seghal posed the following question: IfA andB are rings, doesA[X,X ⁻¹] ℞B[X,X ⁻¹] implyA ℞B. In general the answer to this question is no. In this note we give an affirmative answer in the case thatA andB are Dedekind rings. The author is research assistant at the NFWO.     

Lexicodes over rings

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a \(B\) -ordering over these rings and a selection criterion. As examples we give lexicodes over \(\mathbb Z _4\) and \(\mathbb F _2+u\mathbb F _2\) . It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.     

Polynomial rings over Goldie-Kerr commutative rings II

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).

     

A way to the geometry of rings

This paper is concerned with the problem how to characterize geometrically the affine structure of free modules over rings (associative, with unit-element). The way to solve this problem leads to the concept of the so called affine line-geometry.     

Polynomial rings over commutative reduced Hopfian local rings

We prove that if R is a commutative, reduced, local ring, then R is Hopfian if and only if the ring R[x] is Hopfian. This answers a question of Varadarajan [16], in the case when R is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.     

Elementary equivalence of incidence rings over semi-perfect rings

In this paper, we prove that if two incidence rings constructed by the same semiperfect ring and some two quasi-ordered sets are elementarily equivalent, then the given sets are elementarily equivalent.     

Counting codes over rings

We extend the definition of free codes to codes over local rings and arbitrary Frobenius rings. The number of free codes over finite Frobenius rings is determined by calculating the number for local rings and applying the Chinese Remainder Theorem. A formula for the number of codes of arbitrary type over a finite chain ring is given and this is applied to determine the number of linear codes over a finite principal ideal ring.     

FIm-modules over Noetherian rings

In this paper we study representation theory of the category ${\mathrm{FI}}^m$ introduced in [6], [7] which is a product of copies of the category FI, and show that quite a few interesting representation theoretic and homological properties of FI can be generalized to ${\mathrm{FI}}^m$ in a natural way. In particular, we prove the representation stability property of finitely generated ${\mathrm{FI}}^m$-modules over fields of characteristic 0.     

Modules over serial rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 880–892 June, 1999.     

On skew polynomial rings over locally nilpotent rings

We will show that skew polynomial rings in several variables over locally nilpotent rings cannot contain nonzero idempotent elements. We will also prove that such rings are Brown–McCoy radical.     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.