What is Ring basis?

Question

I'm wondering what ring basis is. Suppose we have a ring $R$, what basis does? For example, suppose we have a cyclotomic ring $R'$, the ring basis is defined as $\{1,\zeta,\ldots, \zeta^{n-1}\}$ where $\zeta$ is a m-th primitive root of unity and $n$ is its degree.

Answer 1

In some situations a ring can have a basis even without being a field. When the ring can be considered as an algebra then the notion basis makes sense. For example the ring $\Bbb Q[x,y]/(x^2+xy+y^2-x-y-2, y^3-y^2-2y+1)$ has as a basis $1, x, x^2, y, xy, x^2y$. It is not a field since with $q_1 = xy-x+1$ and $q_2 = -x^2-2*x*y+y$ we have that $q_1q_2 = (-xy-y^2+x)(x^2+x*y+y^2-x-y-2) + (2x+y)(y^3-y^2-2y+1)$ so the ring has zero divisors and is not a field.

Answer 2

An integral basis for a number ring (a ring of algebraic integers) is a set of numbers such that every element in the ring is a linear combination of elements of the basis with integer coefficients.

A basis like the one you mention in the question is called a power basis.