Affine Algebras: Arithmetic with Ideals

• Construction of ideals and subrings
• Gröbner bases of ideals
• Ideal arithmetic: addition, multiplication, powers, quotients, colon ideals, intersections
• Membership test for ideals
• Test equality and inclusion of ideals

Affine algebras arise commonly in commutative algebra and algebraic geometry. They can also be viewed as generalizations of number fields and algebraic function fields.

If the ideal J of relations defining an affine algebra $A = K[x_1,\ldots,x_n]/J$ is maximal, then A is a field and may be used with any algorithms in Magma which work over fields. Factorization of polynomials over such affine algebras is also supported.

If an affine algebra has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.