6.7 Prime ideals

Another way to write generators of groups of ideals is to use prime ideals. An ideal P of R is prime if for every a and b in R so that ab lies in P at least one of a and b lies in P; an equivalent assertion is that R/P is a domain--non-zero elements give non-zero products. It is clear that the restriction P $\cap$ $\mathbb {Z}$ satisfies the same conditions for integers a and b. In other words P $\cap$ $\mathbb {Z}$ is generated by a prime number p. Thus P determined by the ideal P/pR in R/pR. Now, if R = $\mathbb {Z}$ ^. w₁ + ... + $\mathbb {Z}$ ^. w_n, then R/pR is a vector space of rank n over the finite field $\mathbb {Z}$/p$\mathbb {Z}$. Instead of solving this specific problem, we can ask for a structure theorem for commutative rings with identity with underlying additive group a vector space of rank n over $\mathbb {Z}$/p$\mathbb {Z}$.

Lemma 15   Any ring with underlying additive group a finite dimensional vector space over $\mathbb {Z}$/p$\mathbb {Z}$ is a direct sum of rings of the form ($\mathbb {Z}$/p$\mathbb {Z}$)[T]/(P(T)^m), where P(T) is an irreducible polynomial over the field $\mathbb {Z}$/p$\mathbb {Z}$.

This result follows from the Chinese Remainder theorem and Euclidean division applied to the polynomial ring in one variable ($\mathbb {Z}$/p$\mathbb {Z}$)[T]. If we use the symbol $\mathbb {F}$_q to denote the field with q elements (where q is a prime power), then this result can be refined further as follows

Lemma 16   The ring ($\mathbb {Z}$/p$\mathbb {Z}$)[T]/(P(T)^m), where P(T) is an irreducible polynomial of degree d, is isomorphic to the ring $\mathbb {F}$_p^d[h]/(hⁿ).

This result follows by constructing (using Newton's method of successive approximations) a polynomial $\tilde{T}$ of the form T + P(T)Q₁(T) + ... + P(T)^{n - 1}Q_{n - 1}(T) so that P($\tilde{T}$) is divisible by P(T)ⁿ. Then h = $\tilde{T}$ - T. Combining these results, we see that R/pR has the form

$${\frac{{\mathbb F}_{p^{f_1}}[h_1]}{(h_1^{e_1})}}$$ $$\oplus$$ ^... $$\oplus$$ $${\frac{{\mathbb F}_{p^{f_g}}[h_g]}{(h_g^{e_g})}}$$

Corresponding to each factor we get a surjective ring homomorphism R/pR$\to$$\mathbb {F}$_p^fi. The kernel of this has the form P_i/pR for a prime ideal P_i whose restriction is p. The number f_i is called the residual degree (or more simply the degree) of P_i over p, while the number e_i is called the ramification degree (or order of ramification) of P_i over p.

Now, let I be any ideal. Suppose first that the restriction i of I is of the form i₁i₂, with i₁ and i₂ co-prime. We can apply the Chinese Remainder theorem to write R/I as a direct sum of R/I₁ and R/I₂, where I₁ = I + i₁R and I₂ = I + i₂R. Thus I = I₁ $\cap$ I₂; if I is invertible one easily shows that I = I₁ ^. I₂. Thus we can reduce our study of groups of ideals to the study of ideals Q so that the restriction q is the power of a prime p. Now, R/(Q + pR) is a quotient of the ring studied above and is not zero. Thus there is a prime P_i as above so that Q is contained in P_i. By successively removing such P_i (if Q is invertible) we can write Q as a product of various powers of the P_i considered above. Thus we have written any invertible ideal as a product of (invertible) prime ideals. It is not difficult to show that any such product expression is unique. Thus, we obtain the unique factorisation of ideals in terms of prime ideals.