The Laurent polynomial ring as the principal localisation of Z[t] at t

Let $\mathbb{Z}[t]$ be the polynomial ring over the integers (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring) and let $S=\{t^k:k\ge 0\}\mathrm{.}$ The set $S$ is multiplicative, and the Laurent polynomial ring is the principal localisation ${\mathrm{\Lambda }}_1:=\mathbb{Z}[t{]}_S=S^{-1}\mathbb{Z}[t]$ (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$, Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions). It is a commutative ring with unit $1=t^0$ (Commutative ring). Write again $t$ for the image of the indeterminate under the localisation map; then $t$ is a unit with inverse $t^{-1}$ (A fraction $r\mathrm{/}s$ is a unit in $S^{-1}R$ exactly when $ar\in S$ for some $a\in R$). Every element of ${\mathrm{\Lambda }}_1$ has a representative ${\sum }_{k\in \mathbb{Z}}{a}_k{t}^k,a_k\in \mathbb{Z},$ with only finitely many nonzero coefficients, obtained by writing a fraction $p\mathrm{/}{t}^N$ as a finite $\mathbb{Z}$-linear combination of the powers $t^{k-N}$; the finite coefficient sequence is unique: after multiplying two such sums by a common sufficiently large power of $t$, equality becomes equality of ordinary polynomials. The localisation map $\mathbb{Z}[t]\to {\mathrm{\Lambda }}_1$ is injective because $\mathbb{Z}[t]$ is a domain; hence their coefficients agree (A polynomial ring over an integral domain is an integral domain). Equivalently, two such sums are equal exactly when their coefficient sequences agree (Equality, vanishing, and the kernel of the localisation map).

Universal property. Let $A$ be a ring with unit and let $\phi \text{\hspace{0.17em}⁣}:\mathbb{Z}[t]\to A$ be a unital ring homomorphism with $\phi (t)$ a unit of $A$. Then there is a unique unital ring homomorphism $\tilde{\phi }\text{\hspace{0.17em}⁣}:{\mathrm{\Lambda }}_1\to A$ with $\tilde{\phi }(t)=\phi (t)$. Equivalently, for every unital ring $A$ and every unit $u\in A$ there is a unique unital ring homomorphism ${\mathrm{\Lambda }}_1\to A$ with $t\mapsto u$ (Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

The cited localisation theorem treats commutative target rings. For the possibly noncommutative target used here, define ${\sum }_k{a}_k{t}^k\mapsto {\sum }_k(a_k{1}_A)u^k$. Unique Laurent coefficients make this well defined. Integer multiples of $1_A$ are central, and $u^k{u}^l=u^{k+l}$ for all integers $k,l$, so finite distributivity proves additivity, multiplicativity and preservation of $1$. Every unital homomorphism must send $a_k$ to $a_k{1}_A$ and $t^k$ to $u^k$, proving uniqueness. Taking $u=\phi (t)$ gives the asserted extension of $\phi$.

Augmentation. There is a unique unital ring homomorphism $\epsilon \text{\hspace{0.17em}⁣}:{\mathrm{\Lambda }}_1⟶\mathbb{Z},\epsilon \left({\sum }_{k\in \mathbb{Z}}{a}_k{t}^k\right)={\sum }_{k\in \mathbb{Z}}{a}_k,$ the sum-of-coefficients map (Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$); it is well defined because the coefficients are summable and their total sum is unchanged along the relation of Equality, vanishing, and the kernel of the localisation map. Its kernel is the principal ideal $(t-1)$: indeed $\epsilon (t-1)=0$, and if $\epsilon (x)=0$ then, writing $x={\sum }_{k\in \mathbb{Z}}{a}_k{t}^k$ with finite support and using that ${\sum }_{k\in \mathbb{Z}}{a}_k=0$, one has $x={\sum }_{k\in \mathbb{Z}}{a}_k(t^k-1)=(t-1){\sum }_{k\in \mathbb{Z}}{a}_k(1+t+\cdots +t^{k-1}),$ where $1+t+\cdots +t^{k-1}$ denotes the empty sum $0$ for $k=0$ and, for $k=-j<0$, the negative sum $-(t^{-1}+t^{-2}+\cdots +t^{-j})$, so that $t^{-j}-1=-(t-1)(t^{-1}+\cdots +t^{-j})$; hence $x\in (t-1)$, and the reverse inclusion is $\epsilon (t-1)=0$.

Caveats. This item realises ${\mathrm{\Lambda }}_1$ by localisation and does not identify it with an integral group ring: the additive group underlying ${\mathrm{\Lambda }}_1$ is ${⨁}_{k\in \mathbb{Z}}\mathbb{Z}{t}^k$ and its multiplication is the polynomial one (A polynomial ring over an integral domain is an integral domain is not needed for the ring laws but records that $\mathbb{Z}[t]$ is a domain). The deck-module structures on homology introduced later on this page are defined separately, by the universal property above.