University of the National Education Commission in Krakow
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Papers by Lam Tsit-yuen
Sums of Units in Rings
Journal of Algebra and Its Applications, 2014
In this paper, we study rings that are additively generated by units. We prove that if the identi... more In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units.
ABSTRACT A ring is called clean if every element is the sum of an idempotent and a unit, and a mo... more ABSTRACT A ring is called clean if every element is the sum of an idempotent and a unit, and a module is called clean if its endomorphism ring is a clean ring. In this note we give a short proof of the fact that every continuous or discrete module is clean. The key idea is to show first that every nonsingular injective module is clean by extending the argument for vector spaces.
An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In thi... more An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In this article, we show that, in a matrix ring 𝕄 n (S) (n ≥ 2) over any ring S (with identity), every matrix is a sum of a commutator and a generalized commutator. If S is an elementary divisor ring (in the sense that every square matrix over S is equivalent to a diagonal matrix), then every matrix in 𝕄 n (S) (n ≥ 2) is, in fact, a generalized commutator. Using suitable generic techniques, we show, however, that this conclusion is not true in general for various rings S (e.g. polynomial rings and power series rings). Indeed, for any n ≥ 2, there exists an n × n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator) in 𝕄 n (S). This gives, in particular, a strong negative solution to the problem whether n × n traceless matrices are necessarily commutators over a commutative base ring, for any n ≥ 2.
For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ R a such th... more For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ R a such that 1 − e ∈ (1 − a) R ∩ R (1 − a). A closely related result is that a ring R is an exchange ring if and only if, for every a ∈ R, there exists an idempotent e ∈ R a such that 1 − e ∈ (1 − a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.
ABSTRACT It is well known that every uniquely clean ring is strongly clean. In this article, we i... more ABSTRACT It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules.
In this paper, we establish a determinantal formula for 2 × 2 matrix commutators [X, Y ] = XY − Y... more In this paper, we establish a determinantal formula for 2 × 2 matrix commutators [X, Y ] = XY − Y X over a commutative ring, using (among other invariants) the quantum traces of X and Y . Special forms of this determinantal formula include a "trace version", and a "supertrace version". Some applications of these formulas are given to the study of value sets of binary quadratic forms, the factorization of 2 × 2 integral matrices, and the solution of certain simultaneous diophantine equations over commutative rings.
In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the ... more In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their "strong analogues". The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by developing further hierarchical relationships among these families over special classes of rings, and also by finding concrete examples of ideal families to show that certain such relationships cannot be reversed or improved. The special types of commutative rings studied here include von Neumann regular rings and integral domains (for instance Dedekind domains and valuation domains).
In this paper, we study the class of rings that satisfy internal direct sum cancellation with res... more In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unitregular. Further characterizations for such "IC rings" are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. This approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1. Rings whose matrix rings are IC turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation. We also offer a couple of "hidden" characterizations of unit-regular elements in rings that shed some new light on the relation between similarity and pseudo-similarity-in monoids as well as in rings. The paper concludes with a treatment of ideals for which idempotents lift modulo all 1-sided subideals. An appendix by R.G. Swan 1 on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not IC.
Rings over which the transpose of every invertible matrix is invertible
Journal of Algebra, 2009
We prove that the transpose of every invertible square matrix over a ring R is invertible if and ... more We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if R/rad(R) is commutative. Many other characterizations are obtained for such rings R in terms of U(R) (the group of units of R), including, for instance, c+ba∈U(R)⇒c+ab∈U(R), and 1+abc−cba∈U(R) (for all a,b,c∈R). We also consider a natural weakening of these conditions,
... unit-regular ring. In [CK], Camillo and Khurana proved that every unit-regular ring is clean.... more ... unit-regular ring. In [CK], Camillo and Khurana proved that every unit-regular ring is clean. Recentlyin [KL], Khurana and Lam have shown that a single unit-regular element in a ring may not beclean. As every semisimple ring ...
In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a comm... more In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the "maximal implies prime" variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules.
We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main re... more We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main results also imply that any unit-regular ring has idempotent stable range 1 (and is therefore clean), and that a semilocal ring has idempotent stable range 1 if and only if it is semiperfect.
In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the ... more In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their "strong analogues". The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by developing further hierarchical relationships among these families over special classes of rings, and also by finding concrete examples of ideal families to show that certain such relationships cannot be reversed or improved. The special types of commutative rings studied here include von Neumann regular rings and integral domains (for instance Dedekind domains and valuation domains).
In this paper, we study rings that are additively generated by units. We prove that if the identi... more In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units.
ABSTRACT A ring is called clean if every element is the sum of an idempotent and a unit, and a mo... more ABSTRACT A ring is called clean if every element is the sum of an idempotent and a unit, and a module is called clean if its endomorphism ring is a clean ring. In this note we give a short proof of the fact that every continuous or discrete module is clean. The key idea is to show first that every nonsingular injective module is clean by extending the argument for vector spaces.
An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In thi... more An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In this article, we show that, in a matrix ring 𝕄 n (S) (n ≥ 2) over any ring S (with identity), every matrix is a sum of a commutator and a generalized commutator. If S is an elementary divisor ring (in the sense that every square matrix over S is equivalent to a diagonal matrix), then every matrix in 𝕄 n (S) (n ≥ 2) is, in fact, a generalized commutator. Using suitable generic techniques, we show, however, that this conclusion is not true in general for various rings S (e.g. polynomial rings and power series rings). Indeed, for any n ≥ 2, there exists an n × n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator) in 𝕄 n (S). This gives, in particular, a strong negative solution to the problem whether n × n traceless matrices are necessarily commutators over a commutative base ring, for any n ≥ 2.
For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ R a such th... more For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ R a such that 1 − e ∈ (1 − a) R ∩ R (1 − a). A closely related result is that a ring R is an exchange ring if and only if, for every a ∈ R, there exists an idempotent e ∈ R a such that 1 − e ∈ (1 − a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.
ABSTRACT It is well known that every uniquely clean ring is strongly clean. In this article, we i... more ABSTRACT It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules.
In this paper, we establish a determinantal formula for 2 × 2 matrix commutators [X, Y ] = XY − Y... more In this paper, we establish a determinantal formula for 2 × 2 matrix commutators [X, Y ] = XY − Y X over a commutative ring, using (among other invariants) the quantum traces of X and Y . Special forms of this determinantal formula include a "trace version", and a "supertrace version". Some applications of these formulas are given to the study of value sets of binary quadratic forms, the factorization of 2 × 2 integral matrices, and the solution of certain simultaneous diophantine equations over commutative rings.
In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the ... more In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their "strong analogues". The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by developing further hierarchical relationships among these families over special classes of rings, and also by finding concrete examples of ideal families to show that certain such relationships cannot be reversed or improved. The special types of commutative rings studied here include von Neumann regular rings and integral domains (for instance Dedekind domains and valuation domains).
In this paper, we study the class of rings that satisfy internal direct sum cancellation with res... more In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unitregular. Further characterizations for such "IC rings" are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. This approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1. Rings whose matrix rings are IC turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation. We also offer a couple of "hidden" characterizations of unit-regular elements in rings that shed some new light on the relation between similarity and pseudo-similarity-in monoids as well as in rings. The paper concludes with a treatment of ideals for which idempotents lift modulo all 1-sided subideals. An appendix by R.G. Swan 1 on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not IC.
Rings over which the transpose of every invertible matrix is invertible
Journal of Algebra, 2009
We prove that the transpose of every invertible square matrix over a ring R is invertible if and ... more We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if R/rad(R) is commutative. Many other characterizations are obtained for such rings R in terms of U(R) (the group of units of R), including, for instance, c+ba∈U(R)⇒c+ab∈U(R), and 1+abc−cba∈U(R) (for all a,b,c∈R). We also consider a natural weakening of these conditions,
... unit-regular ring. In [CK], Camillo and Khurana proved that every unit-regular ring is clean.... more ... unit-regular ring. In [CK], Camillo and Khurana proved that every unit-regular ring is clean. Recentlyin [KL], Khurana and Lam have shown that a single unit-regular element in a ring may not beclean. As every semisimple ring ...
In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a comm... more In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the "maximal implies prime" variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules.
We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main re... more We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main results also imply that any unit-regular ring has idempotent stable range 1 (and is therefore clean), and that a semilocal ring has idempotent stable range 1 if and only if it is semiperfect.
In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the ... more In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their "strong analogues". The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by developing further hierarchical relationships among these families over special classes of rings, and also by finding concrete examples of ideal families to show that certain such relationships cannot be reversed or improved. The special types of commutative rings studied here include von Neumann regular rings and integral domains (for instance Dedekind domains and valuation domains).
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Papers by Lam Tsit-yuen