Local ring: Difference between revisions

Revision as of 14:49, 5 August 2013

Revision as of 13:38, 27 November 2013

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* 1 ≠ 0 and the sum of any two non-[[unit (algebra)|unit]]s in ''R'' is a non-unit.

* 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or 1 − ''x'' is a unit.

* If a finite sum is a unit, then soit arehas somea ofterm itsthat termsis a unit (this says in particular that the empty sum is notcannot a be unit, henceso it implies 1 ≠ 0).

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's [[Jacobson radical]]. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,<ref>Lam (2001), p. 295, Thm. 19.1.</ref> necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring ''R'' is local if and only if there do not exist two [[coprime]] proper ([[principal ideal|principal]]) (left) ideals where two ideals ''I''₁, ''I''₂ are called ''coprime'' if ''R'' = ''I''₁ + ''I''₂.