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You can make a polynomial quotient ring:
sage: R.<x> = RR[]
sage: R
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: S = R.quotient(x^6-1)
sage: S
Univariate Quotient Polynomial Ring in xbar over Real Field with 53 bits of precision with modulus x^6 - 1.00000000000000
sage: S.inject_variables()
Defining xbar
Then, you can do
sage: xbar^9 + xbar^11
xbar^5 + xbar^3
sage: xbar^9 * xbar^11
xbar^2
Also, if your coefficients turn out to be rational, i would sugest to define sage: R.<x> = QQ[]
