Ideals in a commutative ring support a rich algebra of operations: sum (I + J), product (IJ), intersection (I ∩ J), and the ideal quotient (I : J) = {r in R : rJ ⊆ I}. These operations interact with prime ideals in precise ways -- for instance, a prime ideal contains an intersection if and only if it contains one of the factors. The radical √I (the set of elements some power of which lies in I) connects ideals to their underlying geometry, with radical ideals corresponding to reduced algebraic sets via the Nullstellensatz.
The four basic operations on ideals -- sum, product, intersection, and quotient -- form the arithmetic backbone of commutative algebra. Given ideals I and J in a commutative ring R, their sum I + J = {a + b : a in I, b in J} is the smallest ideal containing both, their product IJ is the ideal generated by all products ab with a in I, b in J, and their intersection I ∩ J is the largest ideal contained in both. The ideal quotient (or colon ideal) (I : J) = {r in R : rJ ⊆ I} measures how far J is from being contained in I.
These operations satisfy algebraic laws that parallel (but do not exactly replicate) set-theoretic or arithmetic operations. The sum is to ideals what gcd is to integers: in Z, (a) + (b) = (gcd(a, b)). The product is multiplicative: (a)(b) = (ab). The intersection corresponds to lcm: (a) ∩ (b) = (lcm(a, b)). The product always lies inside the intersection (IJ ⊆ I ∩ J), with equality when I and J are comaximal (I + J = R). This last fact is the content of the Chinese Remainder Theorem, which gives an isomorphism R/(I ∩ J) ≅ R/I × R/J when I + J = R.
The radical of an ideal I, denoted √I, is the set {a in R : a^n in I for some n ≥ 1}. It is an ideal containing I, and √I = √(√I) (the radical is idempotent). An ideal equal to its own radical is called a radical ideal. The radical of I equals the intersection of all prime ideals containing I -- this connects the algebraic operation to the prime spectrum. In the special case I = (0), the radical is the nilradical, the intersection of all prime ideals of R. The Jacobson radical (intersection of all maximal ideals) is the other key radical; elements of the Jacobson radical are precisely those r such that 1 - rx is a unit for all x in R.
Interactions between ideal operations and prime ideals are governed by precise rules. A prime ideal contains a product IJ if and only if it contains I or J. A prime ideal contains a finite intersection if and only if it contains one of the factors. These "prime avoidance" properties are the algebraic counterpart of the irreducibility of prime ideals in geometric language, and they underpin primary decomposition, dimension theory, and the theory of associated primes.