A Note on Lifting of Idempotents

In Algebra, it is sometimes useful to construct new algebraic structures from the existing algebraic structures. Often these new structures turn out to be simpler and from these simple structures we can go back to the original structure using some known properties. One such example of this process is studying the factor rings of a ring R and then lifting the properties of the factor ring back to R. The study of lifting idempotents modulo ideals is an important tool to study the structure of rings in this direction. Many people have studied the lifting of idempotents modulo a one-sided ideal R. For instance, It is a classical result that if R is a nil ideal then idempotents can be lifted. Koh [1] gave an elementary proof of this result and also proved that the result is true even for the rings without identity. In literature some classes of rings are even defined in terms of lifting properties. For example, a potent ring R is a ring whose idempotents lift modulo Jacobson radical and every left (equivalently right) ideal which is not contained in the Jacobson Radical contains a non-zero idempotent. Several classes of rings are also characterized by the lifting idempotents modulo ideals.

2. LIFTING IDEMPOTENTS

A right R−module M is said to have the exchange property (see Crawley and Jónsson [2]) if for any module X and decompositions where there exists submodules for each 𝑖 such that . and a module has the finite exchange property if the above condition is met whenever the indexing set I is finite. Warfield in [3] called a ring R an exchange ring, if has the finite exchange property, equivalently, if it has the full exchange property. He proved that that the definition of an exchange ring is left-right symmetric. Local rings, semi regular rings and von Neumann regular rings are all the examples of exchange rings. Nicholson [4] in his classical paper proved that Exchange rings of Warfield can be characterized by lifting properties of idempotents. For this purpose, Nicholson introduced suitable rings.

such that . Nicholson further proved that a ring is suitable ring if idempotents can be lifted modulo every left ideal. Proposition 1.1 (Nicholson [4]) A ring R is suitable iff R/J(R) is suitable and idempotents can be lifted modulo J(R). As for a ring R with identity, end , the following important result by Nicholson proves that Exchange rings are same as suitable rings and hence Exchange rings can be characterized by lifting idempotents modulo ideals. Theorem 1.2 (Nicholson [4]) If R is a ring, the following conditions are equivalent for a left R- module M:

Let R be a ring such that the idempotents lift modulo J(R), where J(R) denotes the Jacobson Radical of ring R. Then implies that there exists such that (see Lemma 5 [5]).

modulo I if, whenever , there exists such that . In the following lemma, they proved that this condition is left-right symmetric even in the case of one-sided ideal.

Even for a commutative ring, Strong lifting is much stronger than ordinary lifting. For example [2] if and , where 𝑝 is a prime number and k is an integer greater than equal to 1, then clearly as the only idempotents in R/I are 0 and 1, idempotents lift modulo I. Taking and , we get . Now 0 cannot lift a, as . So clearly 1 lifts the idempotent and this is not strong lifting as . However, in some cases ordinary lifting can imply strong lifting. The following result is true in this direction.

Strengthening Nicholson‘s result, they proved that exchange rings have in fact strong lifting property modulo each left and right ideal of the ring.

As already discussed, idempotents may not be lifted modulo an arbitrary ideal and even if they are lifted, lifting may not be strong lifting. However, Menal [6] proved the following result:

Further Zhu [7] in 2008 investigated the problem of lifting of central idempotents. An idempotent e is called central if and only if , where is the complementary idempotent of e and we say that a central idempotent can be lifted to R, if there exists a central idempotent such that . For a ring R, a finitely generated projective R-module Zhu associated the problem of lifting central idempotents of a Noetherian (artinian) ring with the rank of groups. For an abelian group 𝐺, the rank of 𝐺, denoted by rank(𝐺) is defined as the dimension of 𝐺⨂ℚ considered as a vector space over ℚ. An ideal J of a ring R is called indecomposable if J is not a direct sum of two nonzero ideals of R. J is called a block ideal of R if J is a direct summand of R and J is indecomposable. A finitely generated projective R − module 𝑃 is called relative power stably free if for each is power stably free as an − module and is power stably free as an − module. He proved the following suitable condition relative to the rank of groups.

A ring R is called semilocal if R/J(R) is a left artinian ring. In the following result, Zhu proved that for semilocal rings, even the converse of above theorem is true.

In this case,

if where denotes the left annihilator of a in R. The ring itself is called a left morphic ring if its every element is left morphic. The following result is proved by Zhu for one-sided morphic rings.

only if the n-by-n full matrix ring over R is an exchange ring. The following results about extensions of Exchange rings are proved by Hong at el. In [8]

And so it follows that property of lifting idempotents modulo every left ideal passes from a ring R to upper (lower) triangular matrix rings, formal power series ring R[[x]] and skew-power series ring R[[x; a]] for some endomorphism a of R. But, in general this property does not pass to subrings. In other words, we can say that subring of an exchange ring is not exchange. For example, ℚ, the field of all rational numbers, is an exchange ring but the subring ℤ, the integer of integers, is not exchange. Petar Pavešić, using the idea of induced lifting, proved that under certain assumptions, subring of an exchange ring is also an exchange ring.

A ring-theoretic property P is called Morita-invariant if, whenever a ring R has the property P, then any ring S Morita-equivalent to R also does. Diesl et. al. in[10] showed that property of lifting idempotents modulo a left ideal is not Morita invariant by proving the following result:

1. 2. , where 3. , where 4. , where 5. , where 6. , where

The problem of lifting idempotents modulo ideals has been studied by many authors in literature and many rings are defined and characterized in terms of lifting properties. In this article, we review the various results in literature about lifting of idempotents and characterization of rings in terms of this property.

1. Koh, Kwangil (1974). "On lifting idempotents." Canadian Mathematical Bulletin 17.4: pp. 607-607. 2. Crawley, Peter, and Bjarni Jónsson (1964). "Refinements for infinite direct decompositions of algebraic systems." Pacific Journal of Mathematics 14.3: pp. 797-855.

4. Nicholson, W. Keith (1977). "Lifting idempotents and exchange rings." Transactions of the American Mathematical Society 229: pp. 269-278. 5. Nicholson, W. K., and Yiqiang Zhou (2005). "Strong lifting." Journal of Algebra 285.2: pp. 795-818. 6. Menal, Pere. "On π-regular rings whose primitive factor rings are artinian." Journal of Pure and Applied Algebra 20.1 (1981): pp. 71-78. 7. Zhu, Xiaosheng (2008). "Lifting Central Idempotents Modulo Jacobson Radicals and Ranks of K 0 Groups." Communications in Algebra 36.8: pp. 2833-2848. 8. Hong, Chan Yong, Nam Kyun Kim, and Yang Lee (2003). "Exchange rings and their extensions." Journal of Pure and Applied Algebra 179.1-2: pp. 117-126. 9. Pavešić, Petar (2010). "Induced liftings, exchange rings and semi-perfect algebras." Journal of Pure and Applied Algebra 214.11: pp. 1901-1906. 10. Diesl, Alexander J., Samuel J. Dittmer, and Pace P. Nielsen (2016). "Idempotent lifting and ring extensions." Journal of Algebra and Its Applications 15.06: pp. 1650112. 11. Khatkar, M. (2017). ―Idempotents of International Journal of Mathematics Trends and Technology. Vol. 46. No. 1: pp. 205-209. 12. Alkan, M. U. S. T. A. F. A., W. K. Nicholson and A. Ç. Özcan (2011). "Strong lifting splits." Journal of pure and applied algebra 215.8: pp. 1879-1888. 13. Warfield, R. B. (1972). ―Exchange rings and decompositions of modules." Mathematische Annalen 199. 1: pp. 31-36.