Integer-valued polynomials on algebras
Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitra...
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| Published in: | Journal of algebra Vol. 373; pp. 414 - 425 |
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| Language: | English |
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01.01.2013
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| ISSN: | 0021-8693, 1090-266X |
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| Abstract | Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine spectrum and Krull dimension of the ring IntD(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in Mn(K), the K-algebra of n×n matrices, that map every matrix in Mn(D) to a matrix in Mn(D). |
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| AbstractList | Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine spectrum and Krull dimension of the ring IntD(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in Mn(K), the K-algebra of n×n matrices, that map every matrix in Mn(D) to a matrix in Mn(D). |
| Author | Frisch, Sophie |
| Author_xml | – sequence: 1 givenname: Sophie surname: Frisch fullname: Frisch, Sophie email: frisch@TUGraz.at organization: Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria |
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| Cites_doi | 10.1007/s00013-005-1225-1 10.1016/j.jnt.2012.05.009 10.1080/00927872.2011.606859 10.1006/jabr.1999.8151 10.1006/jabr.1998.7741 10.1090/S0002-9939-98-04459-1 |
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| Keywords | secondary Krull dimension Polynomial functions Matrix algebras Polynomial rings Integer-valued polynomials Non-commuting variables I-adic topology Non-commutative algebras Polynomial mappings primary Spectrum |
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| References | Frisch (br0100) 2005 Chang (br0070) 1988; 9 Zalesskiĭ (br0140) 1985 Chabert (br0060) 1977; 293/294 Frisch (br0090) 2001; 536 Werner (br0130) 2012; 40 Cahen, Chabert (br0030) 1997; vol. 48 Boynton, Klingler (br0020) 2006; 86 Cahen, Chabert, Frisch (br0040) 2000; 225 Bourbaki (br0010) 1989 Cahen, Chabert, Loper (br0050) 2001; 38 Frisch (br0080) 1999; 211 Loper (br0110) 1998; 126 Loper, Werner (br0120) 2012; 132 Cahen (10.1016/j.jalgebra.2012.10.003_br0050) 2001; 38 Werner (10.1016/j.jalgebra.2012.10.003_br0130) 2012; 40 Loper (10.1016/j.jalgebra.2012.10.003_br0110) 1998; 126 Cahen (10.1016/j.jalgebra.2012.10.003_br0030) 1997; vol. 48 Frisch (10.1016/j.jalgebra.2012.10.003_br0080) 1999; 211 Frisch (10.1016/j.jalgebra.2012.10.003_br0100) 2005 Zalesskiĭ (10.1016/j.jalgebra.2012.10.003_br0140) 1985 Loper (10.1016/j.jalgebra.2012.10.003_br0120) 2012; 132 Chang (10.1016/j.jalgebra.2012.10.003_br0070) 1988; 9 Cahen (10.1016/j.jalgebra.2012.10.003_br0040) 2000; 225 Chabert (10.1016/j.jalgebra.2012.10.003_br0060) 1977; 293/294 Bourbaki (10.1016/j.jalgebra.2012.10.003_br0010) 1989 Boynton (10.1016/j.jalgebra.2012.10.003_br0020) 2006; 86 Frisch (10.1016/j.jalgebra.2012.10.003_br0090) 2001; 536 |
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| Snippet | Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called... |
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| SubjectTerms | I-adic topology Integer-valued polynomials Krull dimension Matrix algebras Non-commutative algebras Non-commuting variables Polynomial functions Polynomial mappings Polynomial rings Spectrum |
| Title | Integer-valued polynomials on algebras |
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