Abstract Nonsense

Crushing one theorem at a time


Point of Post: In this post we discuss the notion of R-algebras where R is some commutative unital ring, and the associated categories.

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Very often when we run into R-modules (where we assume that R is commutative and unital) there is more structure involved, namely the modules are also rings which interact nicely with the R-module structure. Namely, we have a (left, right)R-module A along with a bilinear map \cdot:A\times A\to A which makes A into a unital ring A such that r(x\cdot y)=(rx)\cdot y=x\cdot (ry). We have already run into algebras before, in the context of endomorphism algebras of vector spaces. More generally, any ring of matrices \text{Mat}_n(R) is given the structure of an R-algebra. In fact, it’s easy to see that algebras generalize ring theory since, as we shall see, the category \mathbb{Z}\text{-}\mathbf{Alg} of all \mathbb{Z}-algebras is isomorphic to the category \mathbf{Rng} of rings. Other examples of algebras are polynomial algebras and \mathbb{C} is a 2-dimensional \mathbb{R}-algebra. Algebras play an important role in a lot of algebraic subjects, perhaps most notably with their appearance in differential geometry in the form of tensor algebras, and their appearance in commutative algebra.

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Let R be a commutative unital ring, then we call an R-module (A,+) (no need for distinction about left or right since R is commutative) together with a R-bilinear map \cdot:A\times A\to A such that (A,+,\cdot) is a unital ring, r(x\cdot y)=(rx)\cdot y=x\cdot(ry) and 1_Ra=a for all a\in A to be an R-algebra. 

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There is another common definition of anR-algebra. Namely, let A be a unital ring and let \phi:R\to Z(A) be a unital ring homomorphism, we’d like to consider this the same thing as an R-algebra.

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Indeed, we can define an R-module structure on A by defining r\cdot a=\phi(r)a. To see this, we merely note that

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\begin{aligned}&(r+r')\cdot a=\phi(r+r')a=\phi(r)a+\phi(r')a=ra+ra'\\ &r\cdot(a+a')=\phi(r)(a+a')=\phi(r)a+\phi(r)a'\\ & (rr')a=\phi(rr')a=\phi(r)\phi(r')a=\phi(r)(\phi(r')a)=r(r'a)\end{aligned}

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Moreover, this module structure along with the usual ring structure of A makes A into an R-algebra. The only thing to check is that r(aa')=(ra)a'=a(ra') but this follows since this can be rewritten as \phi(r)(aa')=(\phi(r)a)a'=a(\phi(r)a') which follows from the associativity of A and the fact that \phi(R)\subseteq Z(A). Moreover, the fact that A is unital follows from the fact that 1_Ra=\phi(1_R)a=1_Aa=a for all a\in A from where the conclusion follows.

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Conversely, if A is an R-algebra we can define a map \phi:R\to Z(A) by defining \phi(r)=r\cdot1_A. To see that this is really a unital R-map we note that

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\begin{aligned}&\phi(r+r')=(r+r')1_A=r1_A+r'1_A=\phi(r)+\phi(r')\\ &\phi(rr')=(rr')1_A=(r1_A)(r'1_A)=\phi(r)\phi(r')\\ &\phi(1_R)=1_R1_A=1_A\end{aligned}

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The fact that \phi(R)\subseteq Z(A) follows from the fact that \phi(r)a=(r1_A)a=r(1_Aa)=r(a1_A)=a(r1_A)=a\phi(r) for all a\in A and r\in R.

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To consider some examples, we note that for any commutative unital ring R the polynomial rings R[x_1,\cdots,x_n] and the matrix rings \text{Mat}_n(R). In fact, more generally, given any R-algebra A we have that A[x_1,\cdots,x_n] and \text{Mat}_n(A) are R-algebras with the usual notions of addition and multiplication.

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If X is a topological space then the set C(X) of all continuous maps X\to\mathbb{R} is an \mathbb{R}-algebra, similarly the set C(X;\mathbb{C}) of all continuous maps X\to\mathbb{C} is a \mathbb{C}-algebra.

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Any ring A is a S-algebra in the natural way whenever S\subseteq Z(A).

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The Hamiltonian quaternions \mathbb{H} form a 4-dimensional \mathbb{R}-algebra.

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Let k be some field and V a (not necessarily finitely dimensional) k-space. Then, the endomorphism algebra \text{End}_k(V) is an algebra over k.

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We define a map of R-algebras or R-algebra homomorphism from the R-algebra A to the R-algebra B to be a set function f:A\to B which is both a unital ring homomorphism and an R-map. If we think about the R-algebras A and B as just rings A and B with unital ring homomorphism \phi:R\to A and \psi:R\to B. Then, an R-algebra map f:A\to B is nothing more than a unital ring map f:A\to B such that f\circ\phi=\psi.Indeed, one merely checks that f\circ\phi=\psi implies that

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f(r\cdot x)=f(\phi(r)x)=f(\phi(r))f(x)=\psi(r)f(x)=r\cdot f(x)

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so that f is an R-map as well.

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As an example of an (iso)morphism of R-algebras one can recall the basic fact from linear algebra that if k is a field and V is an n-dimensional k-space then \text{End}_k(V)\cong\text{Mat}_n(k).

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We then define R\text{-}\mathbf{Alg} to be the category of all R-algebras with R-algebra homomorphisms. We define R\text{-}\mathbf{CAlg} to be the full subcategory whose objects are R-algebras A such that A is commutative. It’s fairly easy to see that there is only one way to define a \mathbb{Z}-algebra structure on a given ring A, this is because \mathbb{Z} is initial in \mathbf{Ring} and so there is only one unital homomorphism \mathbb{Z}\to A. Moreover, it’s easy to verify that this is just the usual \mathbb{Z}-module structure on the underlying abelian group of A. Thus, we see that \mathbb{Z}\text{-}\mathbf{Alg}\cong\mathbf{Ring} and \mathbb{Z}\text{-}\mathbf{CAlg}\cong\mathbf{CRing}.

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[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[4] Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.


January 10, 2012 - Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , ,


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