Radical of an Ideal
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called ''radicalization''). The radical of a primary ideal is prime. The radical of an ideal I in a commutative ring R, denoted by Rad(I) or I, is defined as hbox{Rad}(I)={rin R r^nin I hbox...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called ''radicalization''). The radical of a primary ideal is prime. The radical of an ideal I in a commutative ring R, denoted by Rad(I) or I, is defined as hbox{Rad}(I)={rin R r^nin I hbox{for some positive integer} n}. Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Rad(I) turns out to be an ideal itself, containing I. The easiest way to prove that the radical of I of a ring A is an ideal is to note that it is the pre-image of the ideal of nilpotent elements in A / I. In fact, one often takes this identification as a definition of radical. If an ideal I coincides with its own radical, then I is called a radical ideal.
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