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I've always wondered why finitely generated modules are of form

$$M=Ra_1+\dots+Ra_n$$

while finitely generated algebras have form

$$R=k[a_1,\dots, a_n]$$

and finite algebras have form

$$R=ka_1+\dots +ka_n$$

It seems to me that this is an flagrant abuse of nomenclature, and has certainly confused me in the past. Is there some historical reason for this terminology? Any references to the genesis of these terms would be greatly appreciated!

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4 Answers 4

up vote 6 down vote accepted

The terminology is actually very appropriate and precise. Consider that "A is a finitely generated X" means "there exists a finite set G such that A is the smallest X containing G".

Looking at your examples, suppose $M$ is a finitely generated module, generated by $a_1,\dots,a_n$. Then $M$ contains $a_1,\dots,a_n$. Since it is a module, it must contain all elements of the form $Ra_i$ and their sums, so it must contain the module $Ra_1+\dots+Ra_n$. However, since this latter object is in fact a module, $M$ need not contain anything else and is in fact equal to this module.

If $R$ is a finitely generated algebra, we can go through the same procedure as before. However, since algebras have an additional operation (multiplication), we must allow not only sums of elements of the form $ka_n$ but also their products. This gives us that $R$ must contain all polynomial expressions in the elements $a_1,\dots,a_n$, i.e. it must contain the algebra $k[a_1,\dots,a_n]$. Again, since this latter object is in fact an algebra, $R$ need not contain anything else and is equal to this algebra.

A finite algebra seems to be a name for an algebra which is finitely generated as a module. Your example is then consistent with what I wrote above. I do admit that the name seems somewhat misleading.

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+1. I would like to comment that this definition of "finitely generated" is extremely general: it applies not only to rings and modules but to groups, topological groups ("topologically finitely generated"), topological rings, topological modules, $\sigma$-algebras, ... (in the topological examples we need to specify that subobjects should be closed). –  Qiaochu Yuan Aug 20 '12 at 23:04
    
Thanks - I had forgotten about the alternative definition momentarily. I see now that it is completely logical! –  Edward Hughes Aug 21 '12 at 17:25

Note that modules only have one "internal" operation, addition; algebra has two such operations, addition and multiplication.

Closing something under more operations requires more. In particular if you require that $x^n\neq 1$ for all $n$ for the generators.

Observe $\mathbb{N,Z,Q}$. All are generated by the number $1$. Only that $\mathbb N$ requires only closure under addition; $\mathbb Z$ requires also closure under subtraction; and $\mathbb Q$ requires closure under division by non-zero elements.

Similarly if we take $x$ and we add it to a [unital commutative] ring $R$, if we only wish to define $x+x$ and so on, we get a module generated by $x$, that is $R+Rx$. If, on the other hand, we want to allow $x^n$ we effectively get all the polynomials in $x$ (granted we don't want $x$ to be a torsion element), namely $R[x]$.

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I am guessing that OP understands the module vs algebra notation, but finds the finite algebra notation inconsistent with these. I am guessing OP wants to talk about finite algebras. But admittedly, it's not clear to me. –  alex.jordan Aug 20 '12 at 22:37
    
@alex.jordan: Oh. I have never encountered finite algebras in particular (I mean, aside the obvious things)... I'll delete my answer after letting the OP see it and decide whether or not it helps him. –  Asaf Karagila Aug 20 '12 at 22:39
    
Don't delete! Wait and see what OP thinks –  alex.jordan Aug 20 '12 at 22:39
    
@alex: Editing comments is a b***h, and then you reply... :-) –  Asaf Karagila Aug 20 '12 at 22:40
    
Thanks - this is a very helpful answer. Don't delete it, particularly because of the nice examples! –  Edward Hughes Aug 21 '12 at 17:27

A module is an additive structure. Module elements can be added together, not multiplied. So a finitely-generated module should consist of linear combinations of finitely many module elements.

An algebra has both multiplication and addition. The notation for a finitely generated module doesn't communicate that "generators" can be multiplied together. Now there are finitely many algebra elements that are combined in the legal ways (using addition, subtraction, and multiplication by other generators). The new notation $k[a_1,\ldots,a_n]$ tries to communicate this, if only be being something different thatn the module notation.

To me, "finite algebra" is trying to communicate that the algebra is finite dimensional. And we are back to having an additive concept (dimension). So the notation is back to the module style.

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Thank you - I wish I could accept more than one answer because I like this one too! –  Edward Hughes Aug 21 '12 at 17:26

Let's clarify this first: an algebra or module $A$ is finitely generated if there is a finite set $F$ such that $A$ is the smallest subalgebra, respectively submodule, of $A$ that includes $F$. So the definition of finitely generated is the same in both cases. But depending on whether you are dealing with algebras or modules, the finitely generated ones can be written in different forms. Why this is is explained in the other answers.

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