1 | initial version |

You could also work in the quotient ring of of your ring of polynomials by the ideal `(x^3)`

as follows:

```
sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(R.ideal(x^3))
sage: f(x) = x+x^2+x^3+x^4
sage: S(f)
xbar^2 + xbar
sage: S(f).lift()
x^2 + x
```

In the expression for `S(f)`

the variable `xbar`

is the image of `x`

in the quotient ring `S`

.

2 | No.2 Revision |

You could also work in the quotient ring ~~of ~~of your ring of polynomials by the ideal `(x^3)`

as follows:

```
sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(R.ideal(x^3))
sage: f(x) = x+x^2+x^3+x^4
sage: S(f)
xbar^2 + xbar
sage: S(f).lift()
x^2 + x
```

In the expression for `S(f)`

the variable `xbar`

is the image of `x`

in the quotient ring `S`

.

3 | No.3 Revision |

You could also work in the quotient ring of your ring of polynomials by the ideal `(x^3)`

as follows:

```
sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(R.ideal(x^3))
sage: f(x) = x+x^2+x^3+x^4
sage: S(f)
xbar^2 + xbar
sage: S(f).lift()
x^2 + x
```

In the expression for `S(f)`

the variable `xbar`

is the image of `x`

in the quotient ring `S`

.

For another use of the `lift`

method, see this question.

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