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The code in the question defines

• a polynomial ring R and its polynomial variables
• a function of two variables, f, and two symbolic variables

To instead define f as a polynomial from the ring R, use:

sage: R.<x, y> = PolynomialRing(QQ)
sage: f = x + y

Then:

sage: f in R
True

Here is a little more detail.

The instruction

sage: R<x, y> = PolynomialRing(QQ)

is equivalent to

sage: R = PolynomialRing(QQ, ['x', 'y'])
sage: x, y = R.gens()

so it defines R as a polynomial ring whose polynomial variables display as x and y, and it assigns these polynomial variables to the Python variables x and y.

The instruction

sage: f(x, y) = x + y

is equivalent to declaring x and y as symbolic variables, and then declaring f as a symbolic function of these variables.

To check what happens at each step:

sage: preparse("R.<x, y> = PolynomialRing(QQ)")
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = x + y
sage: f
sage: parent(x)
sage: parent(y)
sage: parent(f)
sage: preparse("f(x, y) = x + y")
sage: f(x, y) = x + y
sage: f
sage: parent(x)
sage: parent(y)
sage: parent(f)