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Expositiones Mathematicae

Volume 40, Issue 3, September 2022, Pages 456-468
Expositiones Mathematicae

Unit lifting morphisms

https://doi.org/10.1016/j.exmath.2022.03.003Get rights and content

Abstract

In this paper, we introduce the concept of a unit lifting morphism, a natural generalization of the classical notion “local morphism”, by providing several examples and investigating all its properties and interrelations with local morphisms and unit lifting ideals. Moreover, we expose a relation between a unit lifting morphism f:RS of rings and the pair (ker(f),f1(U(S))) and show that unit lifting morphisms correspond to the least element of a partially ordered set. As a prominent result, we provide the equivalent conditions on the existence of a unit lifting morphism from a ring R into a semisimple artinian ring which is intimately related to a deep result by Camps and Dicks that characterizes semilocal rings in terms of local morphisms. This result gives rise to the study of a new class of rings, which we call weakly semilocal rings.

Introduction

Local morphisms have attracted considerable attention in the last decades, together with the growing interest to the class of local rings, one of the main ingredients for both commutative algebra and algebraic geometry. They are defined as the ring morphisms f:RS, between local commutative rings (R,M) and (S,N), for which f(M)N. As a prominent example of a local morphism one can consider the induced ring maps O(Y,f(x))O(X,x) of a morphism of locally ringed spaces (f,f):(X,OX)(Y,OY) where (X,OX) and (Y,OY) are pairs of topological spaces X,Y and sheaves of rings OX,OY whose stalks are all local rings.

A more general framework of local morphisms was introduced by Cohn in [3]. In the case R,S are not necessarily commutative rings and S is a division ring, he defined a morphism f:RS to be local if it maps non-units to non-units, which is equivalent to saying that for every rR, r is invertible in R whenever f(r) is invertible in S. This definition clearly coincides with the above-mentioned one for R and S commutative local rings. It is easily seen that if a ring R has a local morphism into a division ring, then R is a local ring. The division ring condition placed on S was omitted later by Cohn and the notion of local morphisms reached its present definition. This definition turned into an important tool for understanding semilocal rings (Camps and Dicks [2, Theorem 1]).

In a more recent work [6], Facchini and Herbera studied local morphisms in the noncommutative setting in connection with semilocal endomorphism rings. They showed that local morphisms arise naturally in the construction of the spectral category SpecC of an arbitrary Grothendieck category C.

As mentioned above, reflecting the property of being unit via morphisms is an effective way of understanding many structural properties of not only rings but also other algebraic structures. In many cases, instead of reflecting the property of being unit via morphisms, it is easier to identify structural features of factor rings first and then raise some of the structural features inherent in these factor rings to the ring itself via lifting of units. Let I be an ideal in R. Recall that an element aR is a unit modulo I if there exists an element bR such that 1ab,1baI. In this case, we say that a can be lifted to a unit (modulo I) if there exists a unit uR with auI. Note that the ideal I in R is called unit lifting if, whenever aR is a unit modulo I, then there exists a unit uR with auI. As in the case of local morphisms, lifting of units has prompted continuous interest for many years. In particular, this property was considered by Menal and Moncasi [12] in the study of certain types of self-injective rings; and by Perera [15] in the study of exchange rings and certain classes of C-algebras with real rank zero. In a recent work of Šter [16], it is proved that if I is an ideal of a clean ring R such that R/I is local, then I is unit lifting.

It is therefore natural to seek a new setting for a comprehensive study of both local morphisms and lifting of units via morphisms. In this regard, we introduce the concept of unit lifting morphisms. In the second section, we study this slight general form of local morphisms by investigating its properties and interrelations with not only local morphism but also unit lifting ideals. Moreover, we find an affinity between a unit lifting morphism f:RS and the notions of ker(f) and f1(U(S)) and show that unit lifting morphisms correspond to the least element of a suitable partially ordered set. In Section 3, we prove our main theorem (Theorem 3.2) on the existence of a unit lifting morphism from a ring R into a semisimple artinian ring. This is intimately related to a deep result by Camps and Dicks (see [2, Theorem 1]) that characterizes semilocal rings in terms of local morphisms. Our Main Theorem leads us to consider a natural generalization of semilocal rings, i.e. weakly semilocal rings. We also provide several examples which complement our results and illustrate limitations to the theory. As a final result, a connection between rings with stable range one and weakly semilocal rings is given under a mild assumption.

The rings we consider are associative rings with identity 10, not necessarily commutative. We write J(R) for the Jacobson radical of a ring R, U(R) for the group of units of the ring R, and idem(R) for the set of idempotents. Ring morphisms will be unital, i.e. f(1)=1 for any ring morphism f. A rationally closed subring of R is a subring R such that U(R)=RU(R). This is equivalent to saying that the inclusion map RR is a local morphism. Recall that a ring R is called semilocal if R/J(R) is a semisimple Artinian ring.

Section snippets

Unit lifting morphisms

In this section we study a generalization of local morphisms related to unit lifting ideals; namely, unit lifting morphisms.

Definition 2.1

Let R and S be rings. A ring morphism f:RS is called unit lifting if, for every rR, there exists an invertible element u in R such that f(u)=f(r) whenever f(r) is invertible in S.

Clearly, every local morphism is a unit lifting morphism, but the converse is not generally true. For example, the canonical projection π:ZZ/4Z is clearly unit lifting but not local. More

Weakly semilocal rings

The aim of this section is to prove our Main Theorem 3.2 on the existence of a unit lifting morphism into a semisimple artinian ring. This allows us to define a class of rings, which we call weakly semilocal rings. Due to the above observations between unit lifting morphisms and local morphisms, we recall a result by Rosa Camps and Warren Dicks (see [2, Theorem 1]) that characterizes semilocal rings in terms of local morphisms.

Theorem 3.1

(Camps and Dicks) For a ring R, the following are equivalent:

  • (1)

    R is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank Alberto Facchini (University of Padova) for bringing local morphisms to their attention. The authors are also very grateful to the referee for his/her careful reading and valuable comments.

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