Exercise 24, § 1

Let A be a commutative ring and I the set of all ideals of A. Then $a(b + c) = ab + ac$ for all $a, b, c$ in I, by I, p. 107; in other words I, with the order relation $a \supseteq b$ and the law of composition $(a, b) \mapsto ab$, is a semi-lattice ordered monoid (VI, p. 31, Ex. 13), the sup of two elements a, b of I being $a \cap b$, and their inf $a + b$.

a) Let K be a field and $A = K[X, Y]$ the ring of polynomials in two indeterminates over K. Consider the principal ideals $a = (X), b = (Y), c = (X + Y)$ in A. Show that

b) In the ring A, 1 is a gcd of X and Y, but $(X) + (Y) \neq A$.