- #26

Stephen Tashi

Science Advisor

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to mr. tashi, re post #21: it is true such "substitutions" seem odd, but maybe one can make sense of it by abstracting the ordinarily intuitive process of composition, or substitution.

So saying one can "substitute" an element a of R for the symbol π, is just the fact that one can define a ring homomorphism from Z[π]-->R that sends π to a. I.e. "substituting a for π" just means sending π to a by a ring map. Does this have any (at least logical) appeal?

Yes, it does. Can each homomorphism ##\mathbb{Z}[x] \rightarrow \mathbb{R}## be realized as such a map?

As I understand the ring of polynomials in one "indeterminate" and similar mathematical constructs, they are defined essentially as strings of symbols together with rules for deriving strings from given strings. If done properly, this would be phrased in the same style as is used to define computer languages and in the modeling of formal logic as strings of symbols. I've never seen a algebra text that adopts that style of presentation. In algebra texts it is taken for granted that the reader understands how "substitution" can be used to map a polynomial in one indeterminate to an element in another ring. The concept of "substitution" is so intuitive and universal in mathematics that it isn't formalized. Using the concept of "substitution", we can implement the idea of "composition" of polynomials.

What make me intellectually uneasy is not understanding what, if any, limitations are imposed on the concept of "ring" by assuming it must have a structure that is a string of symbols whose operations are described by manipulating the strings. There is a viewpoint in Group Theory where groups are described in such a manner. I haven't seen this view extended to representing rings.