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Radical of an ideal

A proper ideal typeset structure of typeset structure is called radical of an ideal typeset structure of typeset structure, if typeset structurefor some positive integer typeset structure implies that typeset structure, i.e.

r(a) = {x ∈ R : x^n ∈ a     for    some positive integer n} .

The name goes back to the fact that the radical of an ideal typeset structure is the set of all the possible “roots” of elements of  typeset structure.

The radical of an ideal typeset structure of typeset structure, can be defined also as follows: Consider the projection typeset structure. Then the radical typeset structure is the preimage of  the nilradical typeset structure  ↑  of typeset structure.

Theorem. We have

Theorem. The radical of an ideal typeset structure is the intersection of all prime ideals containing typeset structure.

Theorem. If the radical typeset structure, typeset structure of ideals typeset structure of a ring typeset structure are coprime ↑, then also the ideals typeset structure are coprime.

Cite this web-page as:

Štefan Porubský: Radical.

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