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Moshe Roitman

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Theodore Shifrin

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Boris Kunyavskii

University of Wisconsin-Madison

Anne-Marie Aubert

CNRS - Université Pierre et Marie Curie

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Papers by Moshe Roitman

A complete set of invariants for finite groups and other results

Advances in Mathematics, 1981

up to isomorphism. We also show that under certain assumptions finite groups are determined up to... more up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups. 1. A COMPLETE? SET OF INVARIANTS FOR FINITE GROUPS Let G be a finite group. For given n > 1 and a set Z s { 1,2 ,..., n), we define an element e&Z) of the group ring as follows: E*(g) = g if i E Z, = 1 if i&Z. 301

On unimodular rows

Proceedings of the American Mathematical Society, 1985

We prove here, among other results, that if ( x 0 , … , x n ) ({x_0}, \ldots ,{x_n}) is a unimodu... more We prove here, among other results, that if ( x 0 , … , x n ) ({x_0}, \ldots ,{x_n}) is a unimodular row over a commutative ring A A , n ⩾ 2 n \geqslant 2 , x ∈ A x \in A and \[ x ≡ x n mod J ( A x 0 + ⋯ + A x n − 2 ) x \equiv {x_n}\quad \mod J(A{x_0} + \cdots + A{x_{n - 2}}) \] then ( x 0 , … , x n − 1 , x n ) ∼ E ( x 0 , … , x n − 1 , x ) ({x_0}, \ldots ,{x_{n - 1}},{x_n}){ \sim _E}({x_0}, \ldots ,{x_{n - 1}},x) .

The Kaplansky condition and rings of almost stable range $1$

Proceedings of the American Mathematical Society, 2013

We present some variants of the Kaplansky condition for a K-Hermite ring R R to be an elementary ... more We present some variants of the Kaplansky condition for a K-Hermite ring R R to be an elementary divisor ring. For example, a commutative K-Hermite ring R R is an EDR iff for any elements x , y , z ∈ R x,y,z\in R such that ( x , y ) = R (x,y)=R there exists an element λ ∈ R \lambda \in R such that x + λ y = u v x+\lambda y=uv , where ( u , z ) = ( v , 1 − z ) = R (u,z)=(v,1-z)=R . We present an example of a Bézout domain that is an elementary divisor ring but does not have almost stable range 1 1 , thus answering a question of Warren Wm. McGovern.

Principal Ideal Domains and Euclidean Domains Having 1 as the Only Unit

Communications in Algebra, 2001

ABSTRACT We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal... more ABSTRACT We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F2 or to the polynomial ring F2[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F2 or to F2[X]. We also construct principal ideal domains R of infinite transcendence degree over F2 with the property that 1 is the only unit of R. Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.

$Research paper thumbnail of On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras$

On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras

Cornell University - arXiv, Mar 27, 2022

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach ... more The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Λp1q " 1, is multiplicative, that is, Λpabq " ΛpaqΛpbq for all a, b P A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localisation A P is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra A Ď F X over a subfield F of C, contains all the bounded functions in F X , then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in p0, 8q satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

On finitely stable domains

We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and... more We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and one-dimensional if and only if it is finitely stable and Mori. But we give examples of Archimedean stable local domains that are not one-dimensional. We also prove that a locally Archimedean stable domain satisfies accp and that Archimedean stable semilocal domains are locally Archimedean. But generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

The homogeneous spectrum of a graded commutative ring

Proceedings of the American Mathematical Society, 2001

Suppose Γ \Gamma is a torsion-free cancellative commutative monoid for which the group of quotien... more Suppose Γ \Gamma is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a Γ \Gamma -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose A A is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring A [ Γ ] A[\Gamma ] have Noetherian spectrum. If rank ⁡ Γ ≤ 2 \operatorname {rank} \Gamma \le 2 , we show that A [ Γ ] A[\Gamma ] has Noetherian spectrum, while for each n ≥ 3 n \ge 3 we establish existence of an example where the homogeneous spectrum of A [ Γ ] A[\Gamma ] is not Noetherian.

A characterization of cancellation ideals

Proceedings of the American Mathematical Society, 1997

An ideal I of a commutative ring R with identity is called a cancellation ideal if whenever IB = ... more

On Zsigmondy primes

Proceedings of the American Mathematical Society, 1997

We present simple proofs of Walter Feit's results on large Zsigmondy primes.

Finite generation of powers of ideals

Proceedings of the American Mathematical Society, 1999

Suppose M M is a maximal ideal of a commutative integral domain R R and that some power M n M^n o... more Suppose M M is a maximal ideal of a commutative integral domain R R and that some power M n M^n of M M is finitely generated. We show that M M is finitely generated in each of the following cases: (i) M M is of height one, (ii) R R is integrally closed and ht ⁡ M = 2 \operatorname {ht} M=2 , (iii) R = K [ X ; S ~ ] R = K[X;\tilde S] is a monoid domain over a field K K , where S ~ = S ∪ { 0 } \tilde S = S \cup \{0\} is a cancellative torsion-free monoid such that ⋂ m = 1 ∞ m S = ∅ \bigcap _{m=1}^\infty mS=\emptyset , and M M is the maximal ideal ( X s : s ∈ S ) (X^s:s\in S) . We extend the above results to ideals I I of a reduced ring R R such that R / I R/I is Noetherian. We prove that a reduced ring R R is Noetherian if each prime ideal of R R has a power that is finitely generated. For each d d with 3 ≤ d ≤ ∞ 3 \le d \le \infty , we establish existence of a d d -dimensional integral domain having a nonfinitely generated maximal ideal M M of height d d such that M 2 M^2 is 3 3 -gen...

On finitely stable domains, I

Journal of Commutative Algebra, 2019

We prove that an integral domain R is stable and one-dimensional if and only if R is finitely sta... more We prove that an integral domain R is stable and one-dimensional if and only if R is finitely stable and Mori. If R satisfies these two equivalent conditions, then each overring of R also satisfies these conditions and it is 2-v-generated. We also prove that if R is an Archimedean stable domain such that R is local, then R is one-dimensional and so Mori.

Maximal divisorial ideals and t-maximal ideals

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, ev... more

The minimal number of generators of an invertible ideal

Multiplicative Ideal Theory in Commutative Algebra

When is a linear functional multiplicative?

Transactions of the American Mathematical Society, 1981

We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazk... more We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazko: If φ \varphi is a linear functional on an algebra with unit A A such that φ ( 1 ) = 1 \varphi (1) = 1 and φ ( u ) ≠ 0 \varphi (u) \ne 0 for any invertible u u in A A , then φ \varphi is multiplicative, provided the spectrum of each element in A A is bounded. We present also other conditions which may replace the assumptions on A A in the theorem above.

On stably extended projective modules over polynomial rings

Proceedings of the American Mathematical Society, 1986

We prove here that if A A is a commutative noetherian ring of Krull dimension d d and of finite c... more

The content of a Gaussian polynomial is invertible

Proceedings of the American Mathematical Society, 2005

Let R R be an integral domain and let f ( X ) f(X) be a nonzero polynomial in R [ X ] R[X] . The ... more Let R R be an integral domain and let f ( X ) f(X) be a nonzero polynomial in R [ X ] R[X] . The content of f f is the ideal c ( f ) \mathfrak c(f) generated by the coefficients of f f . The polynomial f ( X ) f(X) is called Gaussian if c ( f g ) = c ( f ) c ( g ) \mathfrak c(fg) = \mathfrak c(f)\mathfrak c(g) for all g ( X ) ∈ R [ X ] g(X) \in R[X] . It is well known that if c ( f ) \mathfrak c(f) is an invertible ideal, then f f is Gaussian. In this note we prove the converse.

On finite generation of powers of ideals

Journal of Pure and Applied Algebra, 2001

Following a joint work with Gilmer and Heinzer, we prove that if M is a maximal ideal of an integ... more Following a joint work with Gilmer and Heinzer, we prove that if M is a maximal ideal of an integral domain R such that some power of M is ÿnitely generated, then M is ÿnitely generated under each of the assumptions below: (a) R is coherent. (b) R is seminormal and M is of height 2. (c) R = K[X ; S] is a monoid domain, M = (X s : s ∈ S), and one of the following conditions holds: • R is seminormal. • ht M = 3 and Q(R) is a ÿnitely generated ÿeld extension of K. For each d ≥ 3 we construct counterexamples of d-dimensional monoid domains as described above.

When is a power series ring n-root closed?

Journal of Pure and Applied Algebra, 1997

Given commutative rings A C B, we present a necessary and sufficient condition for the power seri... more Given commutative rings A C B, we present a necessary and sufficient condition for the power series ring A[[X]] to be n-root closed in B[[X]]. This result leads to a criterion for the the power series ring A[[X]] over an integral domain A to be n-root closed (in its quotient field). For a domain A, we prove: if A is Mori (for example, Noetherian), then A[[X]] is n-root closed iff A is n-root closed; if A is Ptifer, then A[[X]] is root closed iff A is completely integrally closed.

On hilbert functions of reduced and of integral algebras

Journal of Pure and Applied Algebra, 1989

We prove that for any finite field k, there exist differentiable O-sequences which are not Hilber... more We prove that for any finite field k, there exist differentiable O-sequences which are not Hilbert functions of reduced graded k-algebras. We discuss when generic Hilbert functions and the Hilbert function of a complete intersection can be Hilbert functions of reduced or of integral graded algebras. * This work was partially supported by an NSERC grant. M. Roitman thanks Queen's University for its hospitality.

On Nagata's theorem for the class group

Journal of Pure and Applied Algebra, 1990

A complete set of invariants for finite groups and other results

Advances in Mathematics, 1981

up to isomorphism. We also show that under certain assumptions finite groups are determined up to... more up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups. 1. A COMPLETE? SET OF INVARIANTS FOR FINITE GROUPS Let G be a finite group. For given n > 1 and a set Z s { 1,2 ,..., n), we define an element e&Z) of the group ring as follows: E*(g) = g if i E Z, = 1 if i&Z. 301

On unimodular rows

Proceedings of the American Mathematical Society, 1985

We prove here, among other results, that if ( x 0 , … , x n ) ({x_0}, \ldots ,{x_n}) is a unimodu... more We prove here, among other results, that if ( x 0 , … , x n ) ({x_0}, \ldots ,{x_n}) is a unimodular row over a commutative ring A A , n ⩾ 2 n \geqslant 2 , x ∈ A x \in A and \[ x ≡ x n mod J ( A x 0 + ⋯ + A x n − 2 ) x \equiv {x_n}\quad \mod J(A{x_0} + \cdots + A{x_{n - 2}}) \] then ( x 0 , … , x n − 1 , x n ) ∼ E ( x 0 , … , x n − 1 , x ) ({x_0}, \ldots ,{x_{n - 1}},{x_n}){ \sim _E}({x_0}, \ldots ,{x_{n - 1}},x) .

The Kaplansky condition and rings of almost stable range $1$

Proceedings of the American Mathematical Society, 2013

We present some variants of the Kaplansky condition for a K-Hermite ring R R to be an elementary ... more We present some variants of the Kaplansky condition for a K-Hermite ring R R to be an elementary divisor ring. For example, a commutative K-Hermite ring R R is an EDR iff for any elements x , y , z ∈ R x,y,z\in R such that ( x , y ) = R (x,y)=R there exists an element λ ∈ R \lambda \in R such that x + λ y = u v x+\lambda y=uv , where ( u , z ) = ( v , 1 − z ) = R (u,z)=(v,1-z)=R . We present an example of a Bézout domain that is an elementary divisor ring but does not have almost stable range 1 1 , thus answering a question of Warren Wm. McGovern.

Principal Ideal Domains and Euclidean Domains Having 1 as the Only Unit

Communications in Algebra, 2001

ABSTRACT We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal... more ABSTRACT We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F2 or to the polynomial ring F2[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F2 or to F2[X]. We also construct principal ideal domains R of infinite transcendence degree over F2 with the property that 1 is the only unit of R. Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.

$Research paper thumbnail of On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras$

On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras

Cornell University - arXiv, Mar 27, 2022

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach ... more The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Λp1q " 1, is multiplicative, that is, Λpabq " ΛpaqΛpbq for all a, b P A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localisation A P is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra A Ď F X over a subfield F of C, contains all the bounded functions in F X , then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in p0, 8q satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

On finitely stable domains

We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and... more We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and one-dimensional if and only if it is finitely stable and Mori. But we give examples of Archimedean stable local domains that are not one-dimensional. We also prove that a locally Archimedean stable domain satisfies accp and that Archimedean stable semilocal domains are locally Archimedean. But generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

The homogeneous spectrum of a graded commutative ring

Proceedings of the American Mathematical Society, 2001

Suppose Γ \Gamma is a torsion-free cancellative commutative monoid for which the group of quotien... more Suppose Γ \Gamma is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a Γ \Gamma -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose A A is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring A [ Γ ] A[\Gamma ] have Noetherian spectrum. If rank ⁡ Γ ≤ 2 \operatorname {rank} \Gamma \le 2 , we show that A [ Γ ] A[\Gamma ] has Noetherian spectrum, while for each n ≥ 3 n \ge 3 we establish existence of an example where the homogeneous spectrum of A [ Γ ] A[\Gamma ] is not Noetherian.

A characterization of cancellation ideals

Proceedings of the American Mathematical Society, 1997

An ideal I of a commutative ring R with identity is called a cancellation ideal if whenever IB = ... more

On Zsigmondy primes

Proceedings of the American Mathematical Society, 1997

We present simple proofs of Walter Feit's results on large Zsigmondy primes.

Finite generation of powers of ideals

Proceedings of the American Mathematical Society, 1999

Suppose M M is a maximal ideal of a commutative integral domain R R and that some power M n M^n o... more Suppose M M is a maximal ideal of a commutative integral domain R R and that some power M n M^n of M M is finitely generated. We show that M M is finitely generated in each of the following cases: (i) M M is of height one, (ii) R R is integrally closed and ht ⁡ M = 2 \operatorname {ht} M=2 , (iii) R = K [ X ; S ~ ] R = K[X;\tilde S] is a monoid domain over a field K K , where S ~ = S ∪ { 0 } \tilde S = S \cup \{0\} is a cancellative torsion-free monoid such that ⋂ m = 1 ∞ m S = ∅ \bigcap _{m=1}^\infty mS=\emptyset , and M M is the maximal ideal ( X s : s ∈ S ) (X^s:s\in S) . We extend the above results to ideals I I of a reduced ring R R such that R / I R/I is Noetherian. We prove that a reduced ring R R is Noetherian if each prime ideal of R R has a power that is finitely generated. For each d d with 3 ≤ d ≤ ∞ 3 \le d \le \infty , we establish existence of a d d -dimensional integral domain having a nonfinitely generated maximal ideal M M of height d d such that M 2 M^2 is 3 3 -gen...

On finitely stable domains, I

Journal of Commutative Algebra, 2019

We prove that an integral domain R is stable and one-dimensional if and only if R is finitely sta... more We prove that an integral domain R is stable and one-dimensional if and only if R is finitely stable and Mori. If R satisfies these two equivalent conditions, then each overring of R also satisfies these conditions and it is 2-v-generated. We also prove that if R is an Archimedean stable domain such that R is local, then R is one-dimensional and so Mori.

Maximal divisorial ideals and t-maximal ideals

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, ev... more

The minimal number of generators of an invertible ideal

Multiplicative Ideal Theory in Commutative Algebra

When is a linear functional multiplicative?

Transactions of the American Mathematical Society, 1981

We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazk... more We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazko: If φ \varphi is a linear functional on an algebra with unit A A such that φ ( 1 ) = 1 \varphi (1) = 1 and φ ( u ) ≠ 0 \varphi (u) \ne 0 for any invertible u u in A A , then φ \varphi is multiplicative, provided the spectrum of each element in A A is bounded. We present also other conditions which may replace the assumptions on A A in the theorem above.

On stably extended projective modules over polynomial rings

Proceedings of the American Mathematical Society, 1986

We prove here that if A A is a commutative noetherian ring of Krull dimension d d and of finite c... more

The content of a Gaussian polynomial is invertible

Proceedings of the American Mathematical Society, 2005

Let R R be an integral domain and let f ( X ) f(X) be a nonzero polynomial in R [ X ] R[X] . The ... more Let R R be an integral domain and let f ( X ) f(X) be a nonzero polynomial in R [ X ] R[X] . The content of f f is the ideal c ( f ) \mathfrak c(f) generated by the coefficients of f f . The polynomial f ( X ) f(X) is called Gaussian if c ( f g ) = c ( f ) c ( g ) \mathfrak c(fg) = \mathfrak c(f)\mathfrak c(g) for all g ( X ) ∈ R [ X ] g(X) \in R[X] . It is well known that if c ( f ) \mathfrak c(f) is an invertible ideal, then f f is Gaussian. In this note we prove the converse.

On finite generation of powers of ideals

Journal of Pure and Applied Algebra, 2001

Following a joint work with Gilmer and Heinzer, we prove that if M is a maximal ideal of an integ... more Following a joint work with Gilmer and Heinzer, we prove that if M is a maximal ideal of an integral domain R such that some power of M is ÿnitely generated, then M is ÿnitely generated under each of the assumptions below: (a) R is coherent. (b) R is seminormal and M is of height 2. (c) R = K[X ; S] is a monoid domain, M = (X s : s ∈ S), and one of the following conditions holds: • R is seminormal. • ht M = 3 and Q(R) is a ÿnitely generated ÿeld extension of K. For each d ≥ 3 we construct counterexamples of d-dimensional monoid domains as described above.

When is a power series ring n-root closed?

Journal of Pure and Applied Algebra, 1997

Given commutative rings A C B, we present a necessary and sufficient condition for the power seri... more Given commutative rings A C B, we present a necessary and sufficient condition for the power series ring A[[X]] to be n-root closed in B[[X]]. This result leads to a criterion for the the power series ring A[[X]] over an integral domain A to be n-root closed (in its quotient field). For a domain A, we prove: if A is Mori (for example, Noetherian), then A[[X]] is n-root closed iff A is n-root closed; if A is Ptifer, then A[[X]] is root closed iff A is completely integrally closed.

On hilbert functions of reduced and of integral algebras

Journal of Pure and Applied Algebra, 1989

We prove that for any finite field k, there exist differentiable O-sequences which are not Hilber... more We prove that for any finite field k, there exist differentiable O-sequences which are not Hilbert functions of reduced graded k-algebras. We discuss when generic Hilbert functions and the Hilbert function of a complete intersection can be Hilbert functions of reduced or of integral graded algebras. * This work was partially supported by an NSERC grant. M. Roitman thanks Queen's University for its hospitality.

On Nagata's theorem for the class group

Journal of Pure and Applied Algebra, 1990