In algebra, the integral closure of an ideal I {\displaystyle I} of a commutative ring R {\displaystyle R} , denoted by I ¯ {\displaystyle {\overline {I}}} , is the set of all elements r in R {\displaystyle R} that are integral over I {\displaystyle I} : that is, for each i {\displaystyle i} there exists a i ∈ I i {\displaystyle a_{i}\in I^{i}} such that
r n + a 1 r n − 1 + ⋯ + a n − 1 r + a n = 0. {\displaystyle r^{n}+a_{1}r^{n-1}+\cdots +a_{n-1}r+a_{n}=0.}
In other words, r {\displaystyle r} is a zero of a certain kind of monic polynomial. This integral closure is similar to the integral closure of a subring. For example, if R {\displaystyle R} is a domain, an element r in R {\displaystyle R} belongs to I ¯ {\displaystyle {\overline {I}}} if and only if there is a finitely generated R {\displaystyle R} -module M {\displaystyle M} , annihilated only by zero, such that r M ⊆ I M {\displaystyle rM\subseteq IM} . It follows that I ¯ {\displaystyle {\overline {I}}} is an ideal of R {\displaystyle R} (in fact, the integral closure of an ideal is always an ideal; see below). I {\displaystyle I} is said to be integrally closed if I = I ¯ {\displaystyle I={\overline {I}}} . The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
Examples In C [ x , y ] {\displaystyle \mathbb {C} [x,y]} , x i y d − i {\displaystyle x^{i}y^{d-i}} is integral over ( x d , y d ) {\displaystyle (x^{d},y^{d})} . It satisfies the equation r d + ( − x d i y d ( d − i ) ) = 0 {\displaystyle r^{d}+(-x^{di}y^{d(d-i)})=0} , where a d = − x d i y d ( d − i ) {\displaystyle a_{d}=-x^{di}y^{d(d-i)}} is in the d {\displaystyle d} th power of the ideal. Radical ideals (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed. In a normal ring, for any non-zerodivisor x and any ideal I {\displaystyle I} , x I ¯ = x I ¯ {\displaystyle {\overline {xI}}=x{\overline {I}}} . In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed. Let R = k [ X 1 , … , X n ] {\displaystyle R=k[X_{1},\ldots ,X_{n}]} be a polynomial ring over a field k. An ideal I {\displaystyle I} in R {\displaystyle R} is called monomial if it is generated by monomials; i.e., X 1 a 1 ⋯ X n a n {\displaystyle X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}} . The integral closure of a monomial ideal is monomial.
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