[math-fun] Algebra question
Consider the set S of sequences where the nth element is a residue of the nth prime. Natural elementwise addition and multiplication operations can be defined on S. With respect to these operations, S has unique additive and multiplicative identities (0 and 1), as well as additive and multiplicative inverses. S also has a subset isomorphic to the integer, where integer k corresponds to the sequence whose nth element is k modulo the nth prime. Not being a number theorist, I was wondering about the algebraic structure of this set.
It doesn't have multiplicative inverses; what's the inverse of (0, 0, 1, 1, ..,. )? (make the rest of the elements 1). On Tue, Jan 31, 2017 at 5:05 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider the set S of sequences where the nth element is a residue of the nth prime. Natural elementwise addition and multiplication operations can be defined on S. With respect to these operations, S has unique additive and multiplicative identities (0 and 1), as well as additive and multiplicative inverses. S also has a subset isomorphic to the integer, where integer k corresponds to the sequence whose nth element is k modulo the nth prime. Not being a number theorist, I was wondering about the algebraic structure of this set.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
Indeed, it has zero-divisors.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Andy Latto Sent: Tuesday, January 31, 2017 5:18 PM To: math-fun Subject: Re: [math-fun] Algebra question
It doesn't have multiplicative inverses; what's the inverse of (0, 0, 1, 1, ..,. )? (make the rest of the elements 1).
On Tue, Jan 31, 2017 at 5:05 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider the set S of sequences where the nth element is a residue of the nth prime. Natural elementwise addition and multiplication operations can be defined on S. With respect to these operations, S has unique additive and multiplicative identities (0 and 1), as well as additive and multiplicative inverses. S also has a subset isomorphic to the integer, where integer k corresponds to the sequence whose nth element is k modulo the nth prime. Not being a number theorist, I was wondering about the algebraic structure of this set.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Andy Latto -
David Wilson