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I'm not sure how the term is being used here:

Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$.

Given a ring homomorphism $\phi: R \rightarrow R'$ and $x_1, \ldots, x_n \in R'$, there is a unique ring homomorphism $\pi: P \rightarrow R'$ with $\pi\restriction_R = \phi$ and $\pi(X_i) = x_i$ for all $i=1,\ldots,n$. Another way to state this is that $P$ is a universal example of an $R$-algebra with $n$ distinguished elements.

How is it used in general? Also, this example was used as an example of a "universal mapping property" and could you help clarify what this means?

Thank you!

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    See http://en.wikipedia.org/wiki/Universal_property . Universal properties and universal objects are not so easy a thing to describe in words; I think the path to understanding them lies in becoming familiar with enough examples until you've absorbed the underlying idea. Very roughly speaking, they are a strong and surprisingly useful generalization of the idea of a minimal element. Slightly less roughly speaking, at least in some contexts it's reasonable to think of a universal object as the laziest way to accomplish something.2012-09-08
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    @QiaochuYuan "minimal element"? I prefer to think of it as "minimal description" or perhaps "definition that gives you the thing with minimal amount of effort". I think Wikipedia calls it "most efficient construction". But minimal element sounds as if we're talking about a poset. Are we?2012-09-08
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    @Matt: in some sense, yes. Every universal object is an initial or terminal object in a suitable category.2012-09-08
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    @modnar [Here](http://math.stackexchange.com/questions/130850/free-group-and-universal-property) is a thread about the free group, one of the easiest examples of a universal property and [here](http://math.stackexchange.com/questions/132438/universal-properties-again)'s a follow up, still about the free group. There is also [this](http://math.stackexchange.com/questions/130950/free-groups-unique-up-to-unique-isomorphism) and [this](http://math.stackexchange.com/questions/132540/universal-properties-and-diagrams). Hope this helps.2012-09-08

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