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The sequence definition above given in the comment by KConrad is how I first learned to rigorously define R[x] from both Nick Metas and Kenneth Kramer as an undergraduate. Niether worried much about the associativity of product multiplication since it wasn't stated in terms of a free left R-module.

Mario's subsequent definition gives an algebraic structure to R[x] which gives x a very explicit construction-it was also used implicitly by John Terilla in a subsequent ring and module theory course. To me,in mathematics,explicit is good. It seems to be that talking about "the set of symbols" is always something that makes me leery in mathematics because then you get into sticky metaphysics about what you mean by a symbol.(Not that that's strictly bad,mind you-just runs you off topic into quagmires that don't have anything to do with the original question.)

It seems to me Mariano's definition can be made more explicit by replacing "set of symbols" with KConrad's sequences.Essentially, x then becomes a function space of sequences whose range is {0,1} with R[x] then being taken as a free R-module of elements of xx.Note this preserves the idea that the elements of R[x} are not necessarily themselves functions although they are built from a special class of functions i.e. Cauchy sequences of 0's and 1's.

Pietro's definition in the comments above is essentially a special case of this where copies of R are defined in "binary construction" by Cauchy sequences. It has the advantage of being able to give a simple construction of a graded algebra,which of course is quite useful in studying the overall algebra and as a dividend,the polynomial notation drops out trivially.

In fact,the whole thing can be simplified by just assuming the Cauchy construction of R and then using Pietro's definition straight away.

That's my 35 cents on it,anyway.

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The sequence definition above given in the comment by KConrad is how I first learned to rigorously define R[x] from both Nick Metas and Kenneth Kramer as an undergraduate. Niether worried much about the associativity of product multiplication since it wasn't stated in terms of a free left R-module.

Mario's subsequent definition gives an algebraic structure to R[x] which gives x a very explicit construction-it was also used implicitly by John Terilla in a subsequent ring and module theory course. To me,in mathematics,explicit is good. It seems to be that talking about "the set of symbols" is always something that makes me leery in mathematics because then you get into sticky metaphysics about what you mean by a symbol.(Not that that's strictly bad,mind you-just runs you off topic into quagmires that don't have anything to do with the original question.)

It seems to me Mariano's definition can be made more explicit by replacing "set of symbols" with KConrad's sequences.Essentially, x then becomes a function space of sequences whose range is {0,1} with R[x] then being taken as a free R-module of elements of x.

Pietro's definition in the comments above is essentially a special case of this where copies of R are defined in "binary construction" by sequences. It has the advantage of being able to give a simple construction of a graded algebra,which of course is quite useful in studying the overall algebra and as a dividend,the polynomial notation drops out trivially.