Editing Ring (mathematics) (section)

=== Direct product ===
{{Main|Direct product of rings}}

Let {{mvar|R}} and {{mvar|S}} be rings. Then the [[cartesian product|product]] {{math|''R'' × ''S''}} can be equipped with the following natural ring structure:
: <math>\begin{align}
& (r_1,s_1) + (r_2,s_2) = (r_1+r_2,s_1+s_2) \\
& (r_1,s_1) \cdot (r_2,s_2)=(r_1\cdot r_2,s_1\cdot s_2)
\end{align}</math>
for all {{math|''r''{{sub|1}}, ''r''{{sub|2}}}} in {{mvar|R}} and {{math|''s''{{sub|1}}, ''s''{{sub|2}}}} in&nbsp;{{mvar|S}}. The ring {{math|''R'' × ''S''}} with the above operations of addition and multiplication and the multiplicative identity {{math|(1, 1)}} is called the '''[[Direct product of rings|direct product]]''' of {{mvar|R}} with&nbsp;{{mvar|S}}. The same construction also works for an arbitrary family of rings: if {{mvar|R{{sub|i}}}} are rings indexed by a set {{mvar|I}}, then <math display="inline"> \prod_{i \in I} R_i</math> is a ring with componentwise addition and multiplication.

Let {{mvar|R}} be a commutative ring and <math>\mathfrak{a}_1, \cdots, \mathfrak{a}_n</math> be ideals such that <math>\mathfrak{a}_i + \mathfrak{a}_j = (1)</math> whenever {{math|''i'' ≠ ''j''}}. Then the [[Chinese remainder theorem]] says there is a canonical ring isomorphism:
<math display="block">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).</math>

A "finite" direct product may also be viewed as a direct sum of ideals.{{sfnp|Cohn|2003|loc=Theorem 4.5.1|ps=}} Namely, let <math>R_i, 1 \le i \le n</math> be rings, <math display="inline">R_i \to R = \prod R_i</math> the inclusions with the images <math>\mathfrak{a}_i</math> (in particular <math>\mathfrak{a}_i</math> are rings though not subrings). Then <math>\mathfrak{a}_i</math> are ideals of {{mvar|R}} and
<math display="block">R = \mathfrak{a}_1 \oplus \cdots \oplus \mathfrak{a}_n, \quad \mathfrak{a}_i \mathfrak{a}_j = 0, i \ne j, \quad \mathfrak{a}_i^2 \subseteq \mathfrak{a}_i</math>
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&nbsp;{{mvar|R}}. Equivalently, the above can be done through [[central idempotent]]s. Assume that {{mvar|R}} has the above decomposition. Then we can write
<math display="block">1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i.</math>
By the conditions on <math>\mathfrak{a}_i,</math> one has that {{mvar|e{{sub|i}}}} are central idempotents and {{math|1=''e{{sub|i}}e{{sub|j}}'' = 0}}, {{math|''i'' ≠ ''j''}} (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let <math>\mathfrak{a}_i = R e_i,</math> which are two-sided ideals. If each {{mvar|e{{sub|i}}}} is not a sum of orthogonal central idempotents,{{efn|Such a central idempotent is called [[centrally primitive]].}} then their direct sum is isomorphic to&nbsp;{{mvar|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]]).