On the Key Equation Over a Commutative Ring

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton.WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.     

关于强正则环的刻画

通过单边理想是广义弱理想来刻画强正则环,证明了下列条件是等价的:①R是强正则环;②R是半素的左GP-V′-环,且每一个极大的左理想是广义弱理想;③R是半素的左GP-V′-环,且每一个极大的右理想是广义弱理想.     

关于Gpp环和Gp-内射模的若干讨论

讨论了Gpp环和Gp-内射模的一些性质;讨论了Gp-内射模和R-内射模的关系及Gp-内射模和Gpp环的若干联系.最后讨论了和Gpp环有关的有限生成模的投射盖问题.     

FI性质的传递

一个环R称为左(右)FI环，如果它的每一个平坦左(右)R模是内射的，R称为FI环是指它既是左且右的FI环．本讨论了当R是FI环时，其多项式环R[t]，矩阵环MN(R)以及分式不S^-1R也是FI环的充分与必要条件．     

On Almost Armendariz Ring

In this paper,we introduce the notion of an almost Armendariz ring,which is a generalization of an Armendariz ring,and discuss some of its properties.It has bee...     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

主理想整环上n阶矩阵环中的Goldbach问题

Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

形式三角矩阵环上的有限表示模

形式三角矩阵环,又称广义三角矩阵环,这类环及其上的模在环模理论中扮演着重要的角色.本文对形式三角矩阵环上的有限表示模进行了一些探讨.     

关于环上的块循环矩阵环

定义了环R上的块循环矩阵环A，主要证明了下列结论：(1)若J是A的理想，d1，d2，…，dn是R的可逆元，则存在R的理想I使得J=I[σ1，σ2，…，σn]．(2)若d1，d2，…，dn是R的可逆元，则(i)R是单环当且仅当A是单环；(ii)R是局部环当且仅当A是局部环；(iii)J(A)=J(R)[σ1，σ2，…，σn]；(iv)R是半本原环当且仅当A是半本原环．(3)若d1，d2，…，dn都是R的幂零元，则J(A)=J(R) ( (i1,i2,…,im)∈r＼(0,0，…．0n)}RO2 2^1 O2 2^3…O2 2^3.(4)R是左Artin(Noether)环当且仅当A是左Artin(Noether)环．(5)若R有左Morita对偶(自对偶)，则A有左Morita对偶(自对偶)．     

二次整环的剩余类环的性质

主要讨论了二次整环的单位、素元、因子分解、二次整环的剩余类环的性质等问题.得到的主要结果有:二次整环的主理想的特征是无限大;当α满足一定条件时,二次整环关于模(α)的剩余类环是无零因子环,其特征为a2-(uaa,bb)+vb2,并且在一定的限制条件下,剩余类环是一个有限域.关键词:     

有限循环环的构造

给出了T(n)个互不同构的n阶循环环,两个n阶循环环同构的一个充要条件,进一步,得到了n阶循环环的个数是T(n)的一个新证明.     

Pure Injective Modules over a Commutative Valuation Domain

Using geometrical invariants we classify those pure injective modules over a commutative valuation domain which are envelopes of one element.     

本原环为除环的若干条件(英)

本文推广文［1－3］的结果，给出了本原环为除环的几个条件．     

Two Results on Modules whose Endomorphism Ring is Semilocal

The aim of this paper is twofold. On the one hand, we show that the dual Goldie dimension codim(End(M _R )) of the endomorphism ring End(M _R ) of a module M _R can be used as a measure of the dimension of the module M _R . On the other hand, we prove under suitable hypotheses the validity of the Krull–Schmidt Theorem for infinite direct sums of modules with homogeneous semilocal endomorphism rings.     

Morphic环和G-morphic环的一些结果

讨论了morphic环,G-morphic环,PP环,GPP环,Bear环与正则环之间的关系.还证明了在约化环中,强正则环,正则环,π-正则环,G-π-正则环的等价性.     

On Minimal Realization Over a Finite Chain Ring

Let R be a finite chain ring, e.g. a Galois ring. We give a compact recursive formula for a minimal realization of a finite R-sequence. In particular, we show how to obtain a monic minimal polynomial and a rational approximation of a finite R-sequence. We also show how to solve the classical key equation of Algebraic Coding Theory over R.     

On dual of square free modules

A module M is said to be square free if whenever its submodule is isomorphic to N² = N⊕N for some module N, then N = 0. Dually, a module M is said to be d-square free (dual square free) if whenever its factor module is isomorphic to N² for some module N, then N = 0. In this paper, we give some fundamental properties of d-square free modules and study rings whose d-square free modules are closed under submodules or essential extensions.     

关于R—投射模是投射模的环

本文研究了所有Ｒ—投射模都是投射模的环（ＲＰ—环），得出了它的几个等价条件，证明了：Ｓ＝Ｒｎ为ＲＰ—环当且仅当Ｒ为ＲＰ—环；∑ｎｉ＝１Ｒｉ为ＲＰ—环当且仅当每个Ｒｉ为ＲＰ—环．讨论了ＲＰ—环的左投射维数．