Rings, ideals, and characteristic
For a commutative ring \(R\), the polynomial ring \(R[x]\) contains \(R\) as the constant polynomials, and for each \(a\in R\) evaluation \(f\mapsto f(a)\) is a ring morphism \(R[x]\to R\) with kernel \(\gens{x-a}\).
Ring (Tag D-GURUB)
A ring is a triple \((R, +, \cdot)\) consisting of a set \(R\) and binary operations \(+\) and \(\cdot\) on \(R\) such that
\((R, +)\) is an abelian group, with identity \(0\);
\((R, \cdot)\) is a monoid, with identity \(1\);
for all \(a, b, c\in R\), \(a(b+c) = ab + ac\) and \((b+c)a = ba + ca\).
A ring is commutative if additionally \(ab = ba\) for all \(a, b\in R\).
Ring morphism (Tag D-GXMDW)
Let \(R\) and \(S\) be rings. A ring morphism is a map \(f\colon R\to S\) such that for all \(a, b\in R\):
\(f(1_R) = 1_S\);
\(f(a+b) = f(a) + f(b)\);
\(f(ab) = f(a)f(b)\).
Ideal (Tag D-GOFWL)
Let \(R\) be a commutative ring. An ideal \(I\normal R\) is a subset \(I\subseteq R\) such that \((I, +)\) is a subgroup of \((R, +)\) and \(ri \in I\) for all \(r\in R\) and \(i\in I\).
Let \(R\) be a commutative ring. A nonempty subset \(I\subseteq R\) is an ideal if and only if \(I + I \subseteq I\) and \(RI \subseteq I\).
Characteristic (Tag D-JNCUB)
Let \(R\) be a ring. The characteristic of \(R\) is the smallest integer \(n\geq 1\) such that \(\sum_{k=1}^n 1_R = 0\) if such an integer exists, and \(0\) otherwise.
Characteristic of a ring (Tag FD-LXZIW)
Let \(R\) be a ring. If there is an integer \(n\geq 1\) with \(\sum_{j=1}^n 1_R = 0_R\), the characteristic of \(R\) is the smallest such \(n\); otherwise the characteristic of \(R\) is \(0\).
The characteristic of \(R\) is the nonnegative generator \(n\) of the kernel \(n\ZZ\) of the unique ring homomorphism \(\ZZ\to R\): an integer \(m\geq 1\) lies in the kernel exactly when \(\sum_{j=1}^m 1_R=0_R\), and the kernel is \(0\) exactly when no such \(m\) exists.
(Properties inherited by $R[x]$)
If \(R\) is a UFD, then \(R[x]\) is a UFD, and if \(R\) is Noetherian, then \(R[x]\) is Noetherian by the Hilbert basis theorem. If \(R\) is a PID, then \(R[x]\) need not be a PID: the ideal \(\gens{2,x}\normal\ZZ[x]\) is not principal. In fact \(R[x]\) is a PID if and only if \(R\) is a field.
Irreducibility criteria and the field theory of \(k[x]\) for a field \(k\) are on Polynomials over a field.