the phrase "a minimal generator" is used. I don't understand what this means in the absence of a specified set of generators. Can anyone explain
commutative-algebra
asked 2011-12-03
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I think there's a unique minimal set of monomials which generates the ideal. I imagine there's something in Eisenbud's book. – 2011-12-03
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@Dylan: Thanks for the answer, but the phrase is used in various contexts and not just for monomial ideals. This was just an example. – 2011-12-03
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Oh. Do you have another example? – 2011-12-03
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@Dylan: Yes, added. – 2011-12-03
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I’ve seen the term used in the following way in the context of formal languages. Let $\Sigma$ be an alphabet. $L\subseteq\Sigma^*$ is a right ideal if $\varnothing\ne L=L\Sigma^*$, and the minimal generator of $L$ is $L\setminus L\Sigma^+$. This has the same flavor as the definition on p.2 of your second source: ‘If $m\in M-\underline{m}M$, then $m$ is part of a generating set for $M$ of minimum cardinality. In such a case we will say (perhaps abusively) that $m$ is a minimal generator of $M$.’ (Here $\underline{m}$ is the unique maximal ideal of $M$.) – 2011-12-03
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@Brian: Sorry, but I don't quite follow. The authors usually refer to "a minimal generator" of an ideal. I don't understand what you mean by minimal generators of $M$. This phrase is also used with respect to non-maximal ideals. The site does not allow me to add additional links but a quick google search will reveal that. – 2011-12-03
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The quotation tells you exactly what the author means by a minimal generator of $M$, and I know no more than that. I don’t know the answer to your question; I was merely offering a couple of definitions of the term in other contexts in hopes that they would suggest a reasonable interpretation in your context. To me they suggest that in your context a minimal generator of an ideal *might* be any element of $I\setminus I(R\setminus\{1\})$ or something similar. (The maximality of $\overline{m}$ in the second definition is clearly not important to the definition.) – 2011-12-03