The free associative R-algebra on a set and descent of relations

Words and concatenation. Call a word in $S$ a function $w:n\to S$ with $n\in \mathbb{N}$ its length, and write $W:={\bigcup }_{n\in \mathbb{N}}{S}^n$ for the set of words; the empty word is the unique function $0\to S$ (there is exactly one, by [F1]). If $w:m\to S$ and $v:n\to S$, define the concatenation $wv:m+n\to S$ by recursion on $n$: $wv:=w$ when $n=0$; and when $n=\sigma (n^{\mathrm{\prime }})$, write $v^{\mathrm{\prime }}$ for the restriction of $v$ to $n^{\mathrm{\prime }}$, $a:=v(n^{\mathrm{\prime }})$ for the last letter, and define $wv$ as the function $m+\sigma (n^{\mathrm{\prime }})=\sigma (m+n^{\mathrm{\prime }})$ extending $w{v}^{\mathrm{\prime }}$ by the value $a$ at the new index $m+n^{\mathrm{\prime }}$ (the case split uses that every index $i<n$ satisfies $i<n-1$ or $i=n-1$ by trichotomy [F2], and uniqueness of the decomposition $i=m+j$ for $m\le i$ follows from commutativity and cancellation [F2]). Unwinding the two recursive clauses, an index of $wv$ lying in the first $m$ positions carries the corresponding letter of $w$ and an index $i=m+j$ with $j<n$ carries $v(j)$; associativity $(wv)u=w(vu)$ is then proved by induction on the length of $u$: for $u$ of length $0$ both sides are $wv$, and for $u$ with last letter $a$ and predecessor $u^{\mathrm{\prime }}$ one has $w(vu)=w((v{u}^{\mathrm{\prime }})a)=(w(v{u}^{\mathrm{\prime }}))a=((wv)u^{\mathrm{\prime }})a=(wv)u$ by the induction hypothesis and the defining clause, the domains being equal by associativity of addition [F2]; the empty word is neutral since $w\mathrm{\varnothing }=w$ by definition and $\mathrm{\varnothing }v=v$ by the same induction.

The algebra $R⟨S⟩$. Let $F:=R^{(W)}$ be the free $R$-module on the set $W$ of words, with basis $(u_w{)}_{w\in W}$ ([F3]), and let $\mu :F{\otimes }_{R}F\to F$ be the group homomorphism obtained from the $R$-bilinear (hence balanced) pairing $({\sum }_w{a}_w{u}_w,{\sum }_v{b}_v{u}_v)\mapsto {\sum }_{w,v}{a}_w{b}_v{u}_{wv}$ by [F4] and [F9]. Define $fg:=\mu (f\otimes g)$; the displayed formula makes this multiplication $R$-bilinear and gives $u_w{u}_v=u_{wv}$. Its associativity follows from associativity of concatenation of step 1.1 after expanding finite sums by bilinearity, and $u_{\mathrm{\varnothing }}$ is a two-sided identity because $\mathrm{\varnothing }$ is neutral for concatenation; hence $F$ with this multiplication is a unital associative ring, and the map $R\to F$, $r\mapsto r{u}_{\mathrm{\varnothing }}$, is a central unital structure map because scalars commute with the basis, so $R⟨S⟩:=F$ is a unital associative $R$-algebra [F5] with product given on words by concatenation.

Ideal description. Let $R⟨S⟩$ be the $R$-algebra of step 2.1 and let $E\subseteq R⟨S⟩$; let $J$ be the set of finite sums ${\sum }_s{a}_s{e}_s{b}_s$ with $a_s,b_s\in R⟨S⟩$, $e_s\in E$. The empty sum shows $0\in J$; a sum of two such finite sums is again one, and $-aeb=(-a)eb$, so $J$ is an additive subgroup; for $c\in R⟨S⟩$ one has $c(aeb)=(ca)eb$ and $(aeb)c=ae(bc)$, so $J$ is a two-sided ideal [F6]. It contains $E$, since $e=1e1$, and every two-sided ideal $I\supseteq E$ contains every $aeb$ with $e\in E$ by the left and right ideal properties, hence contains all finite sums in $J$; therefore $J$ is a two-sided ideal containing $E$ and contained in every such ideal, so $J=(E)$ by the description of $(E)$ as the intersection [F6].

Universal property. Let $A$ be an $R$-algebra and ${\phi }_0:S\to A$ a map. Define ${\widehat{\phi }}_0(u_w):={\phi }_0(w(0)){\phi }_0(w(1))\cdots {\phi }_0(w(n-1))\in A$ for a word $w:n\to S$ of length $n$, the empty product being $1_A$, and extend $R$-linearly to ${\widehat{\phi }}_0:R⟨S⟩\to A$ by [F3]. Then ${\widehat{\phi }}_0(u_{wv})={\widehat{\phi }}_0(u_w){\widehat{\phi }}_0(u_v)$ by the defining concatenation rule and induction on the length, and ${\widehat{\phi }}_0$ is unital and $R$-linear, so it is a unital $R$-algebra homomorphism [F5] extending ${\phi }_0$ on the words of length one. If $\psi$ is any unital $R$-algebra homomorphism extending ${\phi }_0$, then $\psi (u_w)={\phi }_0(w(0))\cdots {\phi }_0(w(n-1))={\widehat{\phi }}_0(u_w)$ for every word, by multiplicativity and induction on the length, so $\psi ={\widehat{\phi }}_0$ on a spanning set and hence everywhere: the extension is unique.

Quotient universal property. Let $I:=(E)$ and let $\phi :R⟨S⟩\to A$ be an $R$-algebra homomorphism killing $E$, so $E\subseteq \ker \phi$ and hence $I=(E)\subseteq \ker \phi$ by step 3.1 since the kernel of a ring homomorphism is a two-sided ideal. By [F7] there is a unique ring homomorphism $\bar{\phi }:R⟨S⟩\mathrm{/}I\to A$ with $\bar{\phi }(x+I)=\phi (x)$; it is unital and respects the structure maps, hence is an $R$-algebra homomorphism, so it is the unique $R$-algebra factorization, and $R⟨S⟩\mathrm{/}(E)$ is the presented $R$-algebra on generators $S$ and relations $E$.

Tensor algebra. Now let $R=k$ be a field and let $S$ index a basis of a $k$-vector space $V$, so $e:S\to V$, $s\mapsto e_s$ is a bijection onto a basis. The map $S\to T(V)$, $s\mapsto j(e_s)$, extends by step 3.2 to a unital $k$-algebra homomorphism $\mathrm{\Theta }:k⟨S⟩\to T(V)$, and the linear map $V\to k⟨S⟩$ with $e_s\mapsto u_s$ (existing by [F3]) extends by [F8] to a unital $k$-algebra homomorphism $\mathrm{\Xi }:T(V)\to k⟨S⟩$. The composite $\mathrm{\Xi }\mathrm{\Theta }$ fixes every one-letter word $u_s$, hence is the identity by the uniqueness in step 3.2; and $\mathrm{\Theta }\mathrm{\Xi }$ restricts to the identity on $V$, so by the uniqueness in [F8] it is the identity on $T(V)$. Thus $\mathrm{\Theta }$ is an isomorphism of unital $k$-algebras and $k⟨S⟩$ reproduces $T(V)$.

Collecting: step 1.1 constructs the words and their associative concatenation, step 2.1 the $R$-algebra $R⟨S⟩$ free on the words on which the product is concatenation, step 3.2 its universal property, step 4.2 the tensor-algebra identification for $R=k$ a field, step 3.1 the two-sided ideal description and step 4.1 the quotient universal property, so all three parts of the claim hold.