Local rings of rings of quotients M. A. Gomez´ Lozano; M. Siles Molina Departamento de Algebra,´ Geometr´ıay Topolog´ıa, Universidad de M´alaga, 29071 M´alaga, Espa˜na e-mail addresses: [email protected]; [email protected] Abstract: The aim of this paper is to characterize those elements in a semiprime ring R for which taking local algebras and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a. Introduction Local algebras at elements were introduced by K. Meyberg in [15]. These algebras have played a fundamental role in the structure theory of Jordan systems, and they have also proved to be a very useful tool in the setting of associative systems. Local rings at elements make it possible to relate the structure of pairs and triple systems to that of rings, so that the machinery of rings can be applied to these contexts. In [8] local rings at elements were the key tool used to give a Goldie-like theory for associative pairs. In [7] they were a useful device for the study of non-unital exchange rings, which allowed the authors of that paper to show that the exchange property is Morita invariant as well as to extend the exchange property to associative pairs. In [9] the authors presented Fountain-Gould left orders in a different way. Here, the use of local rings at elements made it possible to relate several types of rings of quotients. For instance, [9, Theorem 4.11] related Fountain-Gould left orders and left quotient rings with classical left orders. A connection between Fountain-Gould left orders and classical left orders by using local rings at idempotents was established by P. N. Anh´ and L. M´arkiin [2]. We want to emphasize that “local” properties in a ring R can be characterized by global properties of some of its local rings. For example, the finite left local uniform dimension of a ring is equivalent to the finite left uniform dimension of its local rings. Another example are the semiprime rings with GPI, which are precisely the semiprime rings with at least one PI local ring. This result makes it possible to relate Fountain-Gould left orders with classical Partially supported by the Ministerio de Educaci´ony Ciencia and Fondos Feder, jointly, trough projects MTM2004-03845 and MTM2004-06580-C02-02, and by the Junta de Andaluc´ıa,FQM-264 and FQM336. 1 2 Gomez´ and Siles ones, and will be used here to relate theorems of Amitsur and Martindale with theorems due to Kaplansky and Posner, respectively. In [9, Proposition 3.2(v)] we showed that the local ring Qx at any nonzero element x ∈ R of a left quotient ring Q of a semiprime ring R is again a left quotient ring of Rx, hence it makes sense to compare Qx to the maximal ring of left quotients of Rx. Having in mind this result, the question that must be considered is: for which elements x in R taking local rings at elements and maximal rings of left quotients are commuting operations. A result in this line was given in [5, Corollary 1.9], where it was shown in particular that for a full 2 l ∼ l idempotent e = e in a ring R we have Qmax(eRe) = eQmax(R)e (recall that an idempotent e in a ring R is full if ReR = R). The characterization of the maximal left quotient ring of a ring R without total right zero divisors (Proposition 2.1) and the relation between the dense left ideals of a semiprime ring R and those of its local rings at elements allow us to obtain one of the main results in this paper (Theorem 2.6): 2.6. Theorem Let R be a semiprime ring. For a nonzero element a ∈ R the following conditions are equivalent: l (i) a is von Neumann regular in Qmax(R). l ∼ l (ii) Qmax(Ra) = Qmax(R)a under an isomorphism which is the identity on Ra. The natural question that arises at this point is to characterize those elements of a semiprime ring R that become von Neumann regular in the maximal ring of left quotients Q of R. By Johnson’s Theorem, R is left nonsingular if and only if Q is von Neumann regular, hence Theorem 2.5 for left nonsingular rings implies that taking rings of left quotients and local rings at elements are commuting operations. In particular, a prime ring R containing a PI-element (that is, an element a such that Ra is a PI-ring), is left nonsingular, therefore we may apply the conclusion in the previous l ∼ l paragraph, that is, Qmax(Ra) = Qmax(R)a for every element a ∈ R (Corollary 2.8). From Theorem 2.6 we obtain, as a corollary of Posner’s Theorem, a Martindale like l Theorem, namely, if R is a prime ring with PI-elements, then Qmax(R) is primitive and has nonzero socle. At the end of this section, Theorem 2.10 provides with a necessary and sufficient con- dition for an element a in a ring R to be von Neumannn regular in the maximal ring of left 0 0 quotients of R: the existence of left ideals L and L (of R) such that L⊕lanR(a) and L ⊕La are dense left ideals of R. With the same techniques we obtain in Seccions 3 and 4 results similar to Theorem 2.6 for the Martindale left quotient ring and the maximal symmetric ring of a semiprime ring R. In theses cases, it is not clear which elements in R become von Neumann regular in Q. 1. Local rings at elements. Let a be an element in an arbitrary ring R. The local ring of R at a, denoted by Ra, Local rings of rings of quotients 3 is defined as the ring obtained from the abelian group (aRa, +) by considering the product given by axa · aya = axaya. For example, if e is an idempotent in a ring R, then the local ring of R at e, Re, coincides with the corner eRe (the (1,1)-Pierce component of R relative to this idempotent). This notion turns out to be equivalent to the one established by K. Meyberg in [15] in a Jordan context: Given an element a in the ring R, the a-homotope ring Ra (the abelian group associated to R with product xa·y = xay) has as an ideal the set Ker(a) = {x ∈ R | axa = 0}. Meyberg defined the local ring of R at a as Ra/Ker(a). It is easy to see that the map a Ra → R /Ker(a) defined by axa 7→ x is a ring isomorphism. We will also need an extension of the notion of local ring at an element which we define as follows: Let R be a subring of a ring Q. If b ∈ Q is such that R is a subring of the b homotope Q , then bRb can be regarded as a subring of the local ring Qb of Q at b. This ring will be called the generalized local ring of R at b, and will be denoted by Rb. If b is actually in R, then the definition of generalized local ring at an element agrees, of course, with that given above. 1.1. Lemma. Let R be a prime ring and suppose that for some a ∈ R the ring Ra is left nonsingular. Then R is left nonsingular too. Proof: Suppose 0 6= x ∈ Zl(R), where Zl(R) denotes the left singular ideal of R. By the primeness of R, aRxRa 6= 0. Take 0 6= arxsa, which is in Zl(R) (because the latter is an ideal of R). We have Zl(Rarxsa) = (by [9, Proposition 2.1 (viii)]) Rarxsa = (Ra)arxsa, which must be left nonsingular because the latter is (by [9, Proposition 2.1 (vii)]), a contradiction. Notation. For a left R-homomorphism f : RN → RM, we will write (x)f, or simply xf, to denote the action of f on an arbitrary element x ∈ N. We will denote the action of right R-homomorphisms in a symmetrical way. 2. The maximal left quotient ring. The notion of left quotient ring for a ring without total right zero divisors was introduced by Utumi in 1956 (see [18]). An overring Q of a ring R is said to be a left quotient ring of R if given p, q ∈ Q, with p 6= 0, there exists a ∈ R satisfying ap 6= 0 and aq ∈ R. A right quotient ring is defined analogously. In his paper, Utumi proved that every ring without total right zero divisors (a nonzero element x ∈ R is a total right zero divisor if Rx = 0) has a l unique maximal left quotient ring. This ring, denoted by Qmax(R), is called the maximal left quotient ring of R.