Universal objects: Examples

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 13 March 2012

Quotient groups

Let $N$ be a normal subgroup of $G$.

• The quotient group is the pair $(G/N,\pi )$ where

  $G/N$ is a group       and       $\pi :G\to G/N$ is a group homomorphism

  and the pair $(G/N,\pi )$ is such that
  1. If $K$ is a group and $f:G\to K$ is a homomorphism such that $\ker f\supseteq N$
  2. then there is a unique morphism $\stackrel{\sim }{f}:G/N\to K$ such that

     commutes.

Construction:

A left coset of $N$ in $G$ is a set $$gN=\left\{gn\right|n\in N\},\text{where}g\in G.$$ Define a group with set $$G/N=\left\{\text{cosets}gN\right|g\in G\}$$ and operation $G/N\times G/N\to G/N$ given by $$\left(g_{1}N\right)\left(g_{2}N\right)=g_1{g}_{2}N,$$ and let $\pi :G\to G/N$ be the group homomorphism given by $$\pi \left(g\right)=gN.$$

Quotient rings

Let $R$ be a ring and let $I$ be an ideal of $R$.

• The quotient is the pair $(R/I,\pi )$ where

  $R/I$ is a ring       and       $\pi :R\to R/I$ is a ring homomorphism

  and the pair $(R/I,\pi )$ is such that
  1. If $S$ is a ring and $f:R\to S$ is a ring homomorphism such that $\ker f\supseteq I$
  2. then there is a unique morphism $\stackrel{\sim }{f}:R/I\to S$ such that

     commutes.

Construction:

A coset of $I$ in $R$ is a set $$r+I=\{r+i|i\in I\},\text{where}r\in R.$$ Define a ring with set $$R/I=\{\text{cosets}r+I|r\in R\}$$ and operations $+:R/I\times R/I\to R/I$ and $\cdot :R/I\times R/I\to R/I$ given by $$(r_1+I)+(r_2+I)=r_1+r_2+I\text{and}(r_1+I)(r_2+I)=r_1{r}_2+I,$$ and let $\pi :R\to R/I$ be the ring homomorphism given by $$\pi \left(r\right)=r+I.$$

$(R/I,\pi )$ is the quotient of $R$ by $I$.

Quotient modules

Let $M$ be an $R-$module and let $N$ be an $R-$submodule of $M$.

• The quotient is the pair $(M/N,\pi )$ where

  $M/N$ is an $R-$module       and       $\pi :M\to M/N$ is an $R-$module homomorphism

  and the pair $(M/N,\pi )$ is such that
  1. If $P$ is an $R-$module and $f:M\to P$ is an $R-$module homomorphism such that $\ker f\supseteq N$
  2. then there is a unique morphism $\stackrel{\sim }{f}:M/N\to P$ such that

     commutes.

Construction:

A coset of $N$ in $M$ is a set $$m+N=\{m+n|n\in N\},\text{where}m\in M.$$ Define an $R-$module with set $$M/N=\{\text{cosets}m+N|m\in M\}$$ and operations $+:M/N\times M/N\to M/N$ and $\cdot :R\times M/N\to M/N$ given by $$(m_1+N)+(m_2+N)=m_1+m_2+N\text{and}r\cdot (m+N)=r\cdot m+N.$$ Let $\pi :M\to M/N$ be the $R-$module homomorphism given by $$\pi \left(m\right)=m+N.$$

$(M/N,\pi )$ is the quotient of $M$ by $N$.

Ring of fractions

Let $R$ be a commutative ring and $S$ a subset of $R$.

• The ring of fractions of $R$ with denominators in $S$ is the pair $(Q,\iota )$, where

  $Q$ is a ring       and       $\iota :R\to Q$ is a ring homomorphism

  and the pair $(Q,\iota )$ is such that:
1. If $F$ is a ring and $f:R\to F$ is a ring homomorphism such that all elements of $f\left(S\right)$ are invertible in $F$
2. then there is a unique morphism (of rings) $\tilde{f}:Q\to F$ such that

   commutes.

Construction:

Define a ring with set $Q=\left\{\frac{a}{b}\right|a\in R,b\in S\text{'}\}/=$ where $S\text{'}$ is the multiplicatively closed set containing and generated by $S$, $=$ is the equivalence relation given by

$\frac{a}{b}=\frac{c}{d}$       if       $ads=bcs,$ for some $s\in S\text{'}$,

and with operations given by

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$       and       $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.$

Let $\iota :R\to Q$ be the ring homomorphism given by $$\iota \left(r\right)=\frac{r}{1}.$$

$Q$ is the ring of fractions of $R$ with denominators in $S$.

Group products

Let $(G_i)_i{}_{\in }{}_I$ be a family of groups. The product $\prod _{i\in I}{G}_i$ is the pair $$\left(\prod _{i\in I}{G}_i,({\pi }_i{)}_{i\in I}\right)$$ where

• $\prod _{i\in I}{G}_i$ is a group, and
• $\left({\pi }_i:\prod _{i\in I}{G}_i\to G_i\right)$ is a family of homomorphisms

such that $\left({\prod }_{i\in I}{G}_i,({\pi }_i{)}_{i\in I}\right)$ satisfies

1. if $K$ is a group and $f_i:K\to G_i$ is a family of homomorphisms
2. then there is a unique homomorphism $\tilde{f}:K\to {\prod }_{i\in I}{G}_i$ such that
   commutes for all $i$.

Construction:

Define a group with set $$\prod _{i\in I}{G}_i=\left\{\right(g_i{)}_{i\in I}|g_i\in G_i\}$$ and operation $$\cdot :\prod _{i\in I}{G}_i\times \prod _{i\in I}{G}_i\to \prod _{i\in I}{G}_i$$ given by $$(g_i{)}_{i\in I}\cdot (h_i=\left(g_i{h}_i\right).$$ Define maps ${\pi }_i:\prod _{i\in I}{G}_i\to G_i$ by $${\pi }_i\left(\left(g_j\right)\right)=g_i.$$

$\left(\prod _{i\in I}{G}_i,\left({\pi }_i\right)\right)$ is the product group for the family of groups $(G_i{)}_{i\in I}$.

Restricted products

Let $(G_i{)}_{i\in I}$ be a family of groups. The (restricted) product is the pair $$\left(\prod _{i\in I}{G}_i,\left({\iota }_i\right)\right)$$ where

• $\prod _{i\in I}{G}_i$ is a group, and
• ${\iota }_i:G_i\to \prod _{i\in I}{G}_i$ is a family of homomorphisms

such that

1. if $K$ is a group and $f_i:G_i\to K$ is a family of homomorphisms such that $f_i\left(g\right)$ and $f_j\left(h\right)$ commute for all $i\ne j$, $g\in G_i$, $h\in G_j$
2. then there is a unique morphism $\tilde{f}:\prod _{i\in I}{G}_i\to K$ such that
   commutes for all $i$.

Construction:

Define a group with set $$\prod _{i\in I}{G}_i= {(g_i{)}_{i\in I}|g_i\in G_{i}and\; all\; but\; finitely\; many\; of\; the{g}_{i}are\; equal\; to1}$$ and operation $\cdot :\prod _{i\in I}{G}_i\times \prod _{i\in I}{G}_i\to \prod _{i\in I}{G}_i$ given by $$\left(g_i\right)(g_i\text{'})=(g_i{g}_i\text{'}).$$ Define ${\iota }_i:G_i\to \prod _{i\in I}{G}_i$ by $${\iota }_i\left(g\right)=\left(h_i\right) \; where\; h_j=\{\begin{array}{ll}g,& ifj=i,\\ 1,& j\ne i.\end{array}\phantom{\}}$$

$\left(\prod _{i\in I}{G}_i,{\iota }_i\right)$ is the (restricted) product of the family of groups $(G_i{)}_{i\in I}$.

Inverse limits

Let $I$ be a set with a preordering $\alpha \le \beta$. Let $\left(E_{\alpha },\pi _{\alpha }{}_{\beta }\right)$ be an inverse system of groups. The inverse limit is the pair $$\left(\underleftarrow{\lim }{E}_{\alpha },({\pi }_{\alpha }{)}_{\alpha \in I}\right)$$ where

• $\underleftarrow{\lim }{E}_{\alpha }$ is a group, and
• ${\pi }_{\alpha }:\underleftarrow{\lim }{E}_{\alpha }\to E_{\alpha }$ are morphisms such that ${\pi }_{\alpha }=\pi _{\alpha }{}_{\beta }\circ {\pi }_{\beta }$

and the pair $\left(\underleftarrow{\lim }{E}_{\alpha },({\pi }_{\alpha }{)}_{\alpha \in I}\right)$ is such that

1. if $G$ is a group and $f_{\alpha }:G\to E_{\alpha }$ are morphisms such that $f_{\alpha }=\pi _{\alpha }{}_{\beta }\circ f_{\beta }$
2. then there is a unique morphism $\tilde{f}:G\to \underleftarrow{\lim }{E}_{\alpha }$ such that
   commutes for all $\alpha$.

Construction:

Define a group with set $$\underleftarrow{\lim }{E}_{\alpha }= {\left(g_{\alpha }\right)|\pi _{\alpha }{}_{\beta }\left(g_{\beta }\right)=g_{\alpha }for\alpha \le \beta }$$ and operation $\cdot :\underleftarrow{\lim }{E}_{\alpha }\times \underleftarrow{\lim }{E}_{\alpha }\to \underleftarrow{\lim }{E}_{\alpha }$ given by $$\left(g_{\alpha }\right)\cdot (g_{\alpha }\text{'})=(g_{\alpha }{g}_{\alpha }\text{'}).$$ Define ${\pi }_{\alpha }:\underleftarrow{\lim }{E}_{\alpha }\to E_{\alpha }$ by $${\pi }_{\alpha }\left(\right(g_{\beta }\left)\right)=g_{\alpha }.$$

$\left(\underleftarrow{\lim }{E}_{\alpha },({\pi }_{\alpha }{)}_{\alpha \in I}\right)$ is the inverse limit of the inverse system $\left((E_{\alpha }{)}_{\alpha \in I},\pi _{\alpha }{}_{\beta }\right).$

Direct limits

Let $\left(E_{\alpha },\iota _{\beta }{}_{\alpha }\right)$ be a direct system of groups. The direct limit is the pair $$\left(\underrightarrow{\lim }{E}_{\alpha },{\iota }_{\alpha }\right)$$ where

• $\underrightarrow{\lim }{E}_{\alpha }$ is a group, and
• ${\iota }_{\alpha }:E_{\alpha }\to \underrightarrow{\lim }{E}_{\alpha }$ are morphisms such that ${\iota }_{\beta }\circ \iota _{\beta }{}_{\alpha }={\iota }_{\alpha }$

and $\left(\underrightarrow{\lim }{E}_{\alpha },{\iota }_{\alpha }\right)$ is such that

1. if $G$ is a group and $f_{\alpha }:E_{\alpha }\to G$ are morphisms such that $f_{\beta }\circ \iota _{\beta }{}_{\alpha }=f_{\alpha }$ for all $\alpha$
2. then there is a unique morphism $\tilde{f}:\underrightarrow{\lim }{E}_{\alpha }\to G$ such that
   commutes for all $\alpha$.

Construction:

Define a group with set $$\underrightarrow{\lim }{E}_{\alpha }= {\left(g_{\alpha }\right)|g_{\alpha }\in E_{\alpha },and\; all\; but\; finitely\; many{g}_{\alpha }are1} /~$$ where $~$ is the relation given by $$\left(g_{\alpha }\right)~\left(g_{\beta }\right) \; if\; there\; is\; a\gamma \ge \alpha ,\gamma \ge \beta such\; that\iota _{\gamma }{}_{\alpha }\left(g_{\alpha }\right)=\iota _{\gamma }{}_{\beta }\left(g_{\beta }\right)$$ and operation $\cdot :\underrightarrow{\lim }{E}_{\alpha }\times \underrightarrow{\lim }{E}_{\alpha }\to \underrightarrow{\lim }{E}_{\alpha }$ given by $$\left(g_{\alpha }\right)\cdot \left(g_{\beta }\right)=\left(g_{\alpha }{g}_{\beta }\right).$$ Define ${\iota }_{\alpha }:E_{\alpha }\to \underrightarrow{\lim }{E}_{\alpha }$ by $${\iota }_{\alpha }\left(g\right)=\left(1,...,1,g,1,...\right).$$

$\left(\underrightarrow{\lim }{E}_{\alpha },{\iota }_{\alpha }\right)$ is the direct limit of the direct system $\left(E_{\alpha },{\iota }_{\alpha }\right)$.

Fiber products

Let ${\phi }_1:G_1\to H$ and ${\phi }_2:G_2\to H$ be group homomorphisms. The fiber product is the triplet $$\left(G_1{\times }_H{G}_2,{\pi }_1,{\pi }_2\right)$$ where

• $G_1{\times }_H{G}_2$ is a group, and
• ${\pi }_1:G_1{\times }_H{G}_2\to G_1$ and ${\pi }_2:G_1{\times }_H{G}_2\to G_2$ are homomorphisms such that ${\phi }_1\circ {\pi }_1={\phi }_2\circ {\pi }_2$

and the triplet $\left(G_1{\times }_H{G}_2,{\pi }_1,{\pi }_2\right)$ is such that

1. if $K$ is a group and $f_1:K\to G_1$ and $f_2:K\to G_2$ are homomorphisms such that ${\phi }_1\circ f_1={\phi }_2\circ f_2$
2. then there is a unique morphism $\tilde{f}:K\to G_1{\times }_H{G}_2$ such that
   commutes for each $i$.

Construction:

Define a group with set $$G_1{\times }_H{G}_2= {\left(g_1,g_2\right)\in G_1\times G_2|{\phi }_1\left(g_1\right)={\phi }_2\left(g_2\right)}$$ and operation $\cdot :\left(G_1{\times }_H{G}_2\right)\times \left(G_1{\times }_H{G}_2\right)\to \left(G_1{\times }_H{G}_2\right)$ given by $$\left(g_1,g_2\right)\cdot \left(h_1,h_2\right)=\left(g_1{h}_1,g_2{h}_2\right).$$ Let ${\pi }_1:G_1{\times }_H{G}_2\to G_1$ and ${\pi }_2:G_1{\times }_H{G}_2\to G_2$ be given by $${\pi }_1\left(\left(g_1,g_2\right)\right)=g_1 \; and\; {\pi }_2\left(\left(g_1,g_2\right)\right)=g_2.$$

$\left(G_1{\times }_H{G}_2,{\pi }_1,{\pi }_2\right)$ is the fiber product.

Semidirect products

An extension of $G$ by $F$ is an exact sequence $$1\to F\stackrel{\iota }{\to }E\stackrel{\pi }{\to }G\to 1.$$

A central extension of $G$ by $F$ is an extension of $G$ by $F$ such that $$\iota \left(F\right)\subseteq Z\left(E\right).$$

Let $F$ and $G$ be groups, and $\tau :G\to \mathrm{Aut}\left(F\right)$ a morphism. The semidirect product is the pair $\left(F{\times }_{\tau }G,s\right)$ where

• $1\to F\stackrel{\iota }{\to }F{\times }_{\tau }G\stackrel{\pi }{\to }G\to 1$ is an extension, and
• $s:G\to F{\times }_{\tau }G$ is a section of $\pi$

such that $\left(F{\times }_{\tau }G,s\right)$ satisfies

1. if $1\to F\stackrel{i}{\to }K\stackrel{p}{\to }G\to 1$ is an extension and $s\text{'}:G\to K$ is a section of $p$
2. then there is a unique morphism $\tilde{f}:F{\times }_{\tau }G\to K$ such that
   commutes, and
   commutes.

Construction:

Define a group with set $$F{\times }_{\tau }G= {\left(g,k\right)\in F\times G} =F\times G$$ and operation $\cdot :\left(F{\times }_{\tau }G\right)\times \left(F{\times }_{\tau }G\right)\to F{\times }_{\tau }G$ given by $$\left(g_1,k_1\right)\cdot \left(g_2,k_2\right)=\left(g_1{\tau }_{\left(k_1\right)}\left(g_2\right),k_1{k}_2\right). CHECK\; WITH\; ARUN$$ Define $$\begin{array}{ll}\iota :F\to F{\times }_{\tau }G& by\iota \left(g\right)=\left(g,1\right),\\ \pi :F{\times }_{\tau }G\to G& by\pi \left(\left(g,k\right)\right)=k,and\\ s:G\to F{\times }_{\tau }G& bys\left(k\right)=\left(1,k\right).\end{array}$$

$\left(F{\times }_{\tau }G,s\right)$ is the semidirect product of $F$ with $G$ with respect to $\tau$.

Amalgamated products

Let $(G_i{)}_{i\in I}$ be a family of groups with $A$ as a subgroup. Let $h_i:A\to G_i$ be the injections of $A$ into $G_i$. The amalgamated product is the pair $$\left({*}_A{G}_i,{\iota }_i\right)$$ where

• ${*}_A{G}_i$ is a group, and
• ${\iota }_i:G_i\to {*}_A{G}_i$ are morphisms such that ${\iota }_i{|}_A={\iota }_j{|}_A$ for all $i\ne j$

and the pair $\left({*}_A{G}_i,{\iota }_i\right)$ satisfies

1. if $K$ is a group and $f_i:G_i\to K$ are morphisms such that $f_i{|}_A=f_j{|}_A$ for all $i\ne j$
2. then there is a unique morphism $\tilde{f}:{*}_A{G}_i\to K$ such that
   commutes for all $i\in I$.

Construction:

Define a group with set $${*}_A{G}_i= {\left(g_i\right)|g_i\in G_i} /~$$ where $~$ is the relation given by $$\left(...,1,1,...,1,\underset{i^{\mathrm{th}}}{a},1,...,1\right)~\left(...,1,1,...,1,\underset{j^{\mathrm{th}}}{a},1,...,1\right) \; for\; alli,j$$ and with operation $\cdot :\left({*}_A{G}_i\right)\times \left({*}_A{G}_i\right)\to {*}_A{G}_i$ given by $$\left(g_i\right)(g_i\text{'})=(g_i{g}_i\text{'}).$$ Define ${\iota }_i:G_i\to {*}_A{G}_i$ by $${\iota }_i\left(g\right)=\left(k_j\right) \; where\; k_j=\{\begin{array}{ll}g,& ifj=i,\\ 1,& otherwise.\end{array}\phantom{\}}$$

$\left({*}_A{G}_i,\left({\iota }_i\right)\right)$ is the amalgamated product for the family of groups $(G_i{)}_{i\in I}$.

Products and direct sums of modules

Let $I$ be a set and $(M_i{)}_{i\in I}$ a family of $R-$modules. The product $\prod _{i\in I}{M}_i$ is the pair $$\left(\prod _{i\in I}{M}_i,({\pi }_i{)}_{i\in I}\right)$$ where

• $\prod _{i\in I}{M}_i$ is an $R-$module, and
• $({\pi }_i{)}_{i\in I}$ is a family of $R-$module homomorphisms ${\pi }_i:\prod _{i\in I}{M}_i\to M_i$

such that

1. if $P$ is an $R-$module and $f_i:P\to M_i$ is a family of $R-$module homomorphisms
2. then there is a unique $R-$module homomorphism $\tilde{f}:P\to \prod _{i\in I}{M}_i$ such that
   for all $i\in I$.

The direct sum $\underset{i\in I}{\oplus }{M}_i$ is the pair $$\left(\underset{i\in I}{\oplus }{M}_i,({\iota }_i{)}_{i\in I}\right)$$ where

• $\underset{i\in I}{\oplus }{M}_i$ is an $R-$module, and
• ${\iota }_i:M_i:\underset{i\in I}{\oplus }{M}_i$ is a family of $R-$module homomorphisms

such that

1. if $P$ is an $R-$module and ${\iota }_i:M_i\to \underset{i\in I}{\oplus }{M}_i$ is a family of morphisms
2. then there is a unique morphism $f:\underset{i\in I}{\oplus }{M}_i\to P$ such that
   commutes for all $i$.

Free groups

Let $X$ be a set. The free group on $X$ is the pair $$\left(X^{\pm 1},\iota \right)$$ where

• $X^{\pm 1}$ is a group, and
• $\iota :X\to X^{\pm 1}$ is a map

such that the pair $\left(X^{\pm 1},\iota \right)$ satisfies

1. if $G$ is a group, and $f:X\to G$ is a map
2. then there is a unique group homomorphism $\tilde{f}:X^{\pm 1}\to G$ such that
   commutes.

Construction:

Define a group with set $$X^{\pm 1}= {x_{i_1}^{m_1}\cdots x_{i_{\mathrm{p}}}^{m_{\mathrm{p}}}|x_{i_j}\in X,x_{i_j}\ne x_{i_{j+1}}and{m}_i\in \mathbb{Z}}$$ and with operation given by $$(x_1^{m_1}\cdots x_p^{m_p})(y_1^{1_{}}\cdots y_q^{n_p})=\{\begin{array}{ll}{x}_1^{m_1}\cdots x_{p-1}^{m_{p-1}}{x}_p^{m_p+n_1}{y}_2^{n_2}\cdots y_q^{n_q},& if{x}_p=y_1\\ x_1^{m_1}\cdots x_p^{m_p}{y}_1^{1_{}}\cdots y_q^{n_p},& otherwise.\end{array}\phantom{\}}$$ Define $\iota :X\to X^{\pm 1}$ by $\iota \left(x\right)=x^1.$

$\left(X^{\pm 1},\iota \right)$ is the free group on $X$.

Free modules

Let $X$ be a set. The free module with generating set $X$ is the pair $$\left(R^X,\iota \right)$$ where

• $R^X$ is an $R-$module, and
• $\iota :X\to R^X$ is a map

such that

1. if $M$ is an $R-$module and $f:X\to M$ is a map
2. then there is a unique morphism $\tilde{f}:R^X\to M$ such that
   commutes.

Construction:

Define an $R-$module with set $$R^X= {\sum _{x\in X}{r}_{x}x|r_x\in Rand\; all\; but\; finitely\; many{r}_x=0}$$ and operations $$\begin{array}{rc}\hfill \left(\sum _{x\in X}{r}_{x}x\right)+\left(\sum _{x\in X}{s}_{x}x\right)& =\sum _{x\in X}(r_x+s_x)x, \; and\\ \hfill r\left(\sum _{x\in X}{r}_{x}x\right)& =\sum _x\left(r{r}_x\right)x.\hfill \end{array}$$ Define $\iota :X\to R^X$ by $\iota \left(x\right)=x.$

$\left(R^X,\iota \right)$ is the free module on the set $X$.

Tensor products

Let $M$ be a right $R-$module and $N$ a left $R-$module. The tensor product is the pair $\left(M{\otimes }_{R}N,\pi \right)$ where

• $M{\otimes }_{R}N$ is an abelian group, and
• $\pi :M\times N\to M{\otimes }_{R}N$ is a $\mathbb{Z}-$bilinear map

such that the pair $\left(M{\otimes }_{R}N,\pi \right)$ satisfies

1. if $P$ is an abelian group and $f:M\times N\to P$ is a $\mathbb{Z}-$bilinear map such that $f\left(mr,n\right)=f\left(m,rn\right)$
2. then there is a unique $\mathbb{Z}-$bilinear map $\tilde{f}:M{\otimes }_{R}N\to P$ such that
   commutes.

Construction:

Define a $\mathbb{Z}-$module with set $$M{\otimes }_{R}N= {\sum m_i\otimes n_i|m_i\in M,n_i\in N} /~$$ where $~$ is the equivalence relation determined by $$\begin{array}{rcl}(m_1+m_2)\otimes n& =& m_1\otimes n+m_2\otimes n,\\ m\otimes (n_1+n_2)& =& m\otimes n_1+m\otimes n_2,and\\ mr\otimes n& =& m\otimes rn,\end{array}$$ for all $m\in M,n\in N,r\in R,$ and with operation $$\left(\sum m_i\otimes n_i\right)+\left(\sum m_j\otimes n_j\right)=\sum _{i,j}\left(m_i\otimes n_i+m_j\otimes n_j\right).$$

$\left(M{\otimes }_{R}N,\pi \right)$ is the tensor product of $M$ and $N$.

Algebra of polynomials

Let $I$ be a set. The algebra of polynomials in the set $I$ is the free commutative associative algebra on $I$, i.e. the pair $$\left(R[x_i{]}_{i\in I},\iota \right)$$ where

• $R[x_i{]}_{i\in I}$ is an $R-$algebra, and
• $\iota :I\to R[x_i{]}_{i\in I}$

and such that

1. if $S$ is an $R-$algebra and $f:I\to S$ is a map
2. then there is a unique $R-$algebra homomorphism $\tilde{f}:R[x_i{]}_{i\in I}\to S$ such that
   commutes.

Group algebras

Let $G$ be a group. The group algebra of $G$ (over $R$) is the pair $$\left(R^{\left(G\right)},\iota \right)$$ where

• $R^{\left(G\right)}$ is an $R-$algebra, and
• $\iota :G\to R^{\left(G\right)}$ is a multiplication map

and the pair $\left(R^{\left(G\right)},\iota \right)$ is such that

1. if $F$ is an $R-$algebra and $f:G\to F$ is a multiplicative map
2. then there is a unique $\tilde{f}:R^{\left(G\right)}\to F$ such that
   commutes.

Tensor algebras

The tensor algebra of $M$ is the pair $$\left(T\left(M\right),\iota \right)$$ where

• $T\left(M\right)$ is an $R-$algebra, and
• $\iota :M\to T\left(M\right)$ is an $R-$module homomorphism,

and the pair $\left(T\left(M\right),\iota \right)$ is such that

1. if $E$ is an $R-$algebra and $f:M\to E$ is an $R-$module homomorphism
2. then there is a unique $R-$algebra homomorphism $\tilde{f}:T\left(M\right)\to E$ such that
   commutes.

Free associative algebras

Let $X$ be a set. The free associative algebra is the pair $$\left(R\left[X^{*}\right],\iota \right)$$ where

• $R\left[X^{*}\right]$ is an $R-$algebra, and
• $\iota :X\to R\left[X^{*}\right]$

such that

1. if $F$ is an $R-$algebra and $f:X\to F$
2. then there is a unique $R-$algebra homomorphism $\tilde{f}:R\left[X^{*}\right]\to F$ such that
   commutes.

Notes and References

These notes are from lecture notes of Arun Ram from 2001.

References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

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