<Dl> <Dd> a b: = (a 1 b 1 + ⋯ + a n b n ∣ a i ∈ a and b i ∈ b, i = 1, 2,..., n; for n = 1, 2, ...), (\ displaystyle (\ mathfrak (a)) (\ mathfrak (b)): = \ (a_ (1) b_ (1) + \ dots + a_ (n) b_ (n) \ mid a_ (i) \ in (\ mathfrak (a)) (\ mbox (and)) b_ (i) \ in (\ mathfrak (b)), i = 1, 2, \ dots, n; (\ mbox (for)) n = 1, 2, \ dots \),) </Dd> </Dl> <Dd> a b: = (a 1 b 1 + ⋯ + a n b n ∣ a i ∈ a and b i ∈ b, i = 1, 2,..., n; for n = 1, 2, ...), (\ displaystyle (\ mathfrak (a)) (\ mathfrak (b)): = \ (a_ (1) b_ (1) + \ dots + a_ (n) b_ (n) \ mid a_ (i) \ in (\ mathfrak (a)) (\ mbox (and)) b_ (i) \ in (\ mathfrak (b)), i = 1, 2, \ dots, n; (\ mbox (for)) n = 1, 2, \ dots \),) </Dd> <P> i.e. the product of two ideals a (\ displaystyle (\ mathfrak (a))) and b (\ displaystyle (\ mathfrak (b))) is defined to be the ideal a b (\ displaystyle (\ mathfrak (a)) (\ mathfrak (b))) generated by all products of the form ab with a in a (\ displaystyle (\ mathfrak (a))) and b in b (\ displaystyle (\ mathfrak (b))). The product a b (\ displaystyle (\ mathfrak (a)) (\ mathfrak (b))) is contained in the intersection of a (\ displaystyle (\ mathfrak (a))) and b (\ displaystyle (\ mathfrak (b))). </P> <P> Note that a (\ displaystyle (\ mathfrak (a))) + b (\ displaystyle (\ mathfrak (b))) is also the intersection of all ideals containing both a (\ displaystyle (\ mathfrak (a))) and b (\ displaystyle (\ mathfrak (b))). </P>

Union of two ideals is not an ideal