Is there an accepted notation for the monoid of linear polynomials (with addition as the operation) with coefficients from some ring R?

Like $2p+3$, where $p$ and the identity generate the monoid over the integers?

It's not $Z[p]$, since that would imply a ring where higher order monomials can be present.

And what about multiple indeterminates? Like $2a+3b$, but I don't allow $ab$ in the monoid, as multiplication is not defined.

The examples I've given would be groups if I allowed any integer in the exponent, but the case I'm interested in would only allow positive integers (and zero) in the indeterminate exponents, while any integer would be allowed in the non-indeterminate. So $p-50$ would be in the monoid, but $-p$ would not.

The application I have in mind is a monoid ring that allows linear polynomials in exponents, so I'm working with things like $x^{p-50}$, but I don't do $x^{p^2}$.

  • $\begingroup$ What's the identity? Is it $0$? That wouldn't be a linear polynomial $\endgroup$
    – MCT
    Jun 25 '16 at 3:36
  • $\begingroup$ @Soke, yes identity is 0. So, no, it's not exactly a linear polynomial. Also, the monoid is ordered and I don't allow anything less than 0, so -1 is out. There's probably no notation that exactly fits my monoid, but I was hoping for something close. $\endgroup$ Jun 25 '16 at 3:54
  • $\begingroup$ Do you need the multiplication of the ring structure? It looks like your monoid is the set of polynomials of the form $aX + b$, under addition defined by $(aX + b) + (a'X+b') = (a+a')X + (b+b')$. If this is the case, then your monoid is simply $(R,+) \times (R,+)$. $\endgroup$
    – J.-E. Pin
    Jun 27 '16 at 15:31

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