# What is a generic element in a ring

I am now reading a commutative algebra paper, in which the name "generic element" of a commutative ring appears, however, I can not find the definition in that paper, and also my commutative algebra textbook.

So, could you please tell me what is a generic element in a commutative ring ? Where can I find its definition and related property ? What is it useful for ?

Thank for reading my question !

Edit That paper is a survey on Castelnuovo-Mumford regularity and was written by Le Tuan Hoa. It firstly appear in the lemma 1.3.

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Can you give some context? It may be just a sinonym for 'most elements' or 'all elements'. –  Berci Nov 2 '12 at 15:08
Is there a topology on the ring? –  Dennis Gulko Nov 2 '12 at 15:13
Maybe a non-zero non-unit? –  Your Ad Here Nov 2 '12 at 15:13
There seems to be a specialized meaning of generic element as described in this paper, Integral Closure and Generic Elements. It appears that for ring $R$, a generic element is identified with a linear combination of generators $X_i$ for polynomial ring $R[X_1,..,X_n]$, i.e. a generic point in a multivariate extension of ring $R$. –  hardmath Nov 2 '12 at 15:32
What an unfortunate choice of terminology. Not as bad as defining a special meaning for "arbitrary element," but close! –  rschwieb Nov 2 '12 at 17:33