Baer extensions of rings and stone extensions of semigroups

Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e₁ ε P and e₂ ε Ann(P) such that x→(e₁x, e₂x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed. An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico, Mayaguez, Puerto Rico, March 9–13, 1970.     

The ring of quotients of R(S)

A ring of quotients of the semigroup ring R(S) is discussed where R has a σ-set Σ and S has a σ-set Δ. In particular, we study the cases where (1) R is an integral domain and S is a commutative cancellative semigroup, (2) R is a commutative ring and S is a semilattice and (3) R is a commutative ring and S is a Rees matrix semigroup over a semigroup. Communicated by G. Lallement     

On the artinianness of the local homology modules

In [9] a local homology theory for Artinian modules over commutative rings, which is dual to the local cohomology theory for Noetherian modules, introduced and in [1] the main result of [9] extended. In this note we prove that the local homology modules of an Artinian module over a commutative ring (with identity) are Artinian.     

A note on the units of semigroup rings

In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to extend the results in [2] and 3] to the semigroup ring RG, G an u.p, semigroup.

In the last section we give a necessary and sufficient condi-tion for an element to be a divisor of zero in RG where G is an u.p. semigroup.     

M-McCoy环和M-Armendariz环的多项式扩张

研究非交换环上的相对于幺半群的McCoy环和Armendariz环的多项式扩张.对于包含无限循环子幺半群的交换可消幺半群M,证明了若R是M-McCoy（或M-Armendariz）环,则R上的洛朗多项式环R[x,x-1]是M-McCoy（或M-Armendariz）环.     

Commutativity Conditions for Rings and Generalized 2-CN Rings

Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions: (1) if for each x ∈ R\N(R) and each y ∈ R,(xy)k =xkyk for k =m,m + 1,n,n + 1,where m and n are relatively prime positive integers,then R is commutative;(2) if for each x ∈ R\J(R) and each y ∈ R,(xy)k =ykxk for k =m,m+ 1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

On semigroups admitting ring structure

A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero. Communicated by A. H. Clifford     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

环的投射生成元和K0群

设Ｒ是含幺结合环,Ｐｇ（Ｒ）是Ｒ的所有投射生成元的同构类组成的半群,Ｇｒ（Ｐｇ（Ｒ））是Ｐｇ（Ｒ）的Ｇｒｏｔｈｅｎｄｉｅｃｋ群,在本文中我们证明了Ｋ０（Ｒ）＝Ｇｒ（Ｐｇ（Ｒ））。由此我们得到对任意ＶＢＮ环Ｒ,存在环Ｓ满足Ｓ＾２＝Ｓ并且具有Ａｕｔ－Ｐｉｃ性质,最后我们给出了环的一个分类,并且用Ｐｇ（Ｒ）的周期性对它作了描述。     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

Commutativity of One Sided S- Unital Rings

In the present paper we extend some commutativity theorems for rings as follows: Let m > 1, n and k be fixed non- negative integers, and let R be a left or right s- unital ring satisfying the polynomial identity [xⁿ]y ? y^mx^k,x] = 0. Then R is commutative. Under appropriate conditions the commutativity of R has also been proved for the case m = 1.     

关于半准素环

交换环Ｒ称为（受限制的）半准素环，如果对Ｒ的每个（非零）主理想Ａ，都有A^1/2是Ｒ的素理想，本文刻画了受限制的半准素环，给出了有单位元的Ｎｏｅｔｈｅｒ受限制的半准素环的分类以及半准素整环是伪赋值整环的一个条件     

Analogue of G. Higman's Theorem for Commutative Torsion-Free Rings

In this paper we consider finite rank torsion-free rings, which have almost regular automorphisms of prime order (a non-trivial automorphism is called almost regular if it has only trivial fixed points, i.e. zero and the elements of a ring linear dependent on its identity). The main result of this paper is the analogue of G. Higman's known Theorem [1] on almost regular automorphism for commutative finite rank torsion-free rings.     

Rings with x(yz)=z(yx)

In non-associative rings the associative law is replaced by various weaker identities. In this paper the identity x.yz = z.yx is considered. Let R be a ring satisfying this identity. It is obvious that if R has an identity element, then R is both commutative and associative. It is shown that if R is prime, third power associative, 2-torsion free, and has a nonzero idempotent, then this idempotent must be an identity and hence R must be commutative and associative.     

Polynomial Rings over Symmetric Rings Need Not Be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

Multiplicative Semigroup of a Residue Class Ring

In this paper, we consider the structure of the multiplicative semigroup of a residue class ring R/I of a commutative ring R with identity modulo its nonzero ideal I. For the general case, we investigate the H-classes, maximal subgroups and the structure of Reg(R/I) which is the set of regular elements of R/I. If R is any integral domain and if I is a product of powers of invertible maximal ideals, we show that R/I is an epigroup, every H^*-class of R/I is a nil-extension of a group (:unipotent epigroup) and that R/I is a complete lattice of unipotent epigroups.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.