Birings and plethories of integer-valued polynomials

Jesse Elliott[1]

  • [1] Department of Mathematics California State University, Channel Islands Camarillo, California 93012

Actes des rencontres du CIRM (2010)

  • Volume: 2, Issue: 2, page 53-58
  • ISSN: 2105-0597

Abstract

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Let and be commutative rings with identity. An --biring is an -algebra together with a lift of the functor from -algebras to sets to a functor from -algebras to -algebras. An -plethory is a monoid object in the monoidal category, equipped with the composition product, of --birings. The polynomial ring is an initial object in the category of such structures. The -algebra has such a structure if is a domain such that the natural -algebra homomorphism is an isomorphism for and injective for . This holds in particular if is an isomorphism for all , which in turn holds, for example, if is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor from -algebras to -algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.

How to cite

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Elliott, Jesse. "Birings and plethories of integer-valued polynomials." Actes des rencontres du CIRM 2.2 (2010): 53-58. <http://eudml.org/doc/196274>.

@article{Elliott2010,
abstract = {Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor $\operatorname\{Hom\}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A[X]$ is an initial object in the category of such structures. The $D$-algebra $\{\operatorname\{Int\}\}(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta _n: \{\bigotimes _D\}_\{i = 1\}^n \{\operatorname\{Int\}\}(D) \rightarrow \{\operatorname\{Int\}\}(D^n)$ is an isomorphism for $n = 2$ and injective for $n \le 4$. This holds in particular if $\theta _n$ is an isomorphism for all $n$, which in turn holds, for example, if $D$ is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor $\operatorname\{Hom\}_D(\{\operatorname\{Int\}\}(D),-)$ from $D$-algebras to $D$-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.},
affiliation = {Department of Mathematics California State University, Channel Islands Camarillo, California 93012},
author = {Elliott, Jesse},
journal = {Actes des rencontres du CIRM},
keywords = {Biring; plethory; integer-valued polynomial},
language = {eng},
number = {2},
pages = {53-58},
publisher = {CIRM},
title = {Birings and plethories of integer-valued polynomials},
url = {http://eudml.org/doc/196274},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Elliott, Jesse
TI - Birings and plethories of integer-valued polynomials
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 53
EP - 58
AB - Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor $\operatorname{Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A[X]$ is an initial object in the category of such structures. The $D$-algebra ${\operatorname{Int}}(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta _n: {\bigotimes _D}_{i = 1}^n {\operatorname{Int}}(D) \rightarrow {\operatorname{Int}}(D^n)$ is an isomorphism for $n = 2$ and injective for $n \le 4$. This holds in particular if $\theta _n$ is an isomorphism for all $n$, which in turn holds, for example, if $D$ is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor $\operatorname{Hom}_D({\operatorname{Int}}(D),-)$ from $D$-algebras to $D$-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.
LA - eng
KW - Biring; plethory; integer-valued polynomial
UR - http://eudml.org/doc/196274
ER -

References

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  4. J. Elliott, Binomial rings, integer-valued polynomials, and -rings, J. Pure Appl. Alg. 207 (2006) 165–185. Zbl1100.13026MR2244389
  5. J. Elliott, Universal properties of integer-valued polynomial rings, J. Algebra 318 (2007) 68–92. Zbl1129.13022MR2363125
  6. J. Elliott, Some new approaches to integer-valued polynomial rings, in Commutative Algebra and its Applications: Proceedings of the Fifth Interational Fez Conference on Commutative Algebra and Applications, Eds. Fontana, Kabbaj, Olberding, and Swanson, de Gruyter, New York, 2009. Zbl1177.13053MR2606288
  7. J. Elliott, Biring and plethory structures on integer-valued polynomial rings, to be submitted for publication. 
  8. C. J. Hwang and G. W. Chang, Bull. Korean Math. Soc. 35 (2) (1998) 259–268. Zbl0917.13002MR1623691
  9. D. O. Tall and G. C. Wraith, Representable functors and operations on rings, Proc. London Math Soc. (3) 20 (1970) 619–643. Zbl0226.13007MR265348

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