information:highlighting, structural:expansion
Direct product

Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:

for all r, r in R and s, s in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the direct product of R with S. The same construction also works for an arbitrary family of rings: if R are rings indexed by a set I, then ∏ i ∈ I R i {\textstyle \prod _{i\in I}R_{i}} is a ring with componentwise addition and multiplication.

Let R be a commutative ring and a 1, ⋯, a n {\displaystyle {\mathfrak {a}}_{1},\cdots,{\mathfrak {a}}_{n}} be ideals such that a i + a j = (1) {\displaystyle {\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)} whenever i ≠ j. Then the Chinese remainder theorem says there is a canonical ring isomorphism: R / ⋂ i = 1 n a i ≃ ∏ i = 1 n R / a i, x mod ⋂ i = 1 n a i ↦ (x mod a 1, …, x mod a n). {\displaystyle R/{\textstyle \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod {\textstyle \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots,x{\bmod {\mathfrak {a}}}_{n}).}

A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let R i, 1 ≤ i ≤ n {\displaystyle R_{i},1\leq i\leq n} be rings, R i → R = ∏ R i {\textstyle R_{i}\to R=\prod R_{i}} the inclusions with the images a i {\displaystyle {\mathfrak {a}}_{i}} (in particular a i {\displaystyle {\mathfrak {a}}_{i}} are rings though not subrings). Then a i {\displaystyle {\mathfrak {a}}_{i}} are ideals of R and R = a 1 ⊕ ⋯ ⊕ a n, a i a j = 0, i ≠ j, a i 2 ⊆ a i {\displaystyle R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}} as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central idempotents. Assume that R has the above decomposition. Then we can write 1 = e 1 + ⋯ + e n, e i ∈ a i. {\displaystyle 1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.} By the conditions on a i, {\displaystyle {\mathfrak {a}}_{i},} one has that e are central idempotents and e e = 0, i ≠ j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let a i = R e i, {\displaystyle {\mathfrak {a}}_{i}=Re_{i},} which are two-sided ideals. If each e is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to R.

An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).