= Solution
For a <compact Lie group> $G$, the complex <representation ring of a compact group> $R(G)$ is the <Grothendieck group> of finite-dimensional <continuous> complex <group representations>, with the relations $[M\oplus N]=[M]+[N]$. Multiplication is $[M][N]=[M\otimes N]$, and the unit is the trivial one-dimensional <representation>. The ring of <class functions> is
$$
c\ell(G)=\{f\in C(G,\mathbb C):f(hgh^{-1})=f(g)
\text{ for all }g,h\in G\},
$$
with pointwise addition and multiplication. The <character> of $M$ is $\chi_M(g)=\operatorname{tr}\rho_M(g)$. The <trace> identities for direct sums and tensor products make
$$
[M]-[N]\longmapsto\chi_M-\chi_N
$$
a well-defined <ring homomorphism> $\chi:R(G)\to c\ell(G)$.
Normalize <Haar measure> by $\int_Gdg=1$. Averaging a positive definite <Hermitian inner product> gives
$$
(v,w)_G=\int_G(\rho(g)v,\rho(g)w)_0\,dg,
$$
which is positive definite and $G$-invariant. This proves <unitarization of a compact-group representation>. In a <unitary representation> the <orthogonal complement> of an <invariant subspace> is invariant. Induction on dimension therefore gives <complete reducibility of compact-group representations>. Hence every element of $R(G)$ is a finite integral linear combination of irreducible classes.
We establish the needed <character orthogonality for compact groups> carefully. For irreducible unitary <representations> $V,W$, let $G$ act on $\operatorname{Hom}_{\mathbb C}(W,V)$ by
$$
g\cdot A=\rho_V(g)A\rho_W(g)^{-1}.
$$
Its average $P=\int_G(g\cdot)\,dg$ is a projection onto $\operatorname{Hom}_G(W,V)$: averaging makes every image invariant, and it fixes every intertwiner. The <trace> of this action is $\chi_V(g)\overline{\chi_W(g)}$, since the inverse of a unitary <matrix> has conjugate <trace>. Taking the <trace> of the projection gives
$$
\int_G\chi_V(g)\overline{\chi_W(g)}\,dg
=\dim\operatorname{Hom}_G(W,V).
$$
For completeness, <Schur's lemma> follows here from invariance of the kernel and image of an intertwiner. A nonzero intertwiner between irreducibles is an isomorphism. An endomorphism of an irreducible complex <representation> has an <eigenvalue> $\lambda$; the noninvertible intertwiner $A-\lambda I$ must vanish. Thus the last dimension is one for isomorphic irreducibles and zero for inequivalent ones.
If a virtual <character> $\sum_Vn_V\chi_V$ vanishes, integrating it against $\overline{\chi_W}$ gives $n_W=0$ for every irreducible $W$. The irreducible multiplicities therefore distinguish the classes, and
$$
\boxed{\chi:R(G)\longrightarrow c\ell(G)\text{ is injective}.}
$$
This also proves that two finite-dimensional <representations> with the same <character> are isomorphic.
For <SU(2)>, the irreducibles are $V_k=\operatorname{Sym}^k(\mathbb C^2)$, $k\geq0$, of dimension $k+1$. One can see this classification through the usual $\mathfrak{sl}_2$ operators $H,E,F$: a <highest-weight vector> $v$ has $Hv=kv$, $Ev=0$, and the relations $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$ give, by induction,
$$
EF^jv=j(k-j+1)F^{j-1}v.
$$
The torus weights are integers. If $F^rv\ne0$ but $F^{r+1}v=0$, the identity at $j=r+1$ forces $k=r\geq0$. The vectors $v,Fv,\ldots,F^kv$ span an invariant irreducible module, which in an irreducible <representation> is the whole space. This is the symmetric-power module. Conversely its successive monomial weight vectors are linked by $E,F$ with nonzero coefficients, proving its irreducibility. Thus the <classification of finite-dimensional representations of SU2> gives every irreducible, rather than just a list of examples.
On the <maximal torus> $T=\{\operatorname{diag}(z,z^{-1}):|z|=1\}$, their <characters> are
$$
\chi_k(z)=z^k+z^{k-2}+\cdots+z^{-k}.
$$
With $x=\chi_1=z+z^{-1}$, multiplication gives $\chi_{k+1}=x\chi_k-\chi_{k-1}$, starting with $\chi_0=1$. Hence each $\chi_k$ is a monic integral polynomial of degree $k$ in $x$. For example $\chi_2=x^2-1$ and $\chi_3=x^3-2x$. They form a basis over $\mathbb Z$, so
$$
\boxed{R(SU(2))\cong\mathbb Z[x],\qquad x=[\mathbb C^2].}
$$
The <character> map realizes this ring as the finite integral polynomials in $z+z^{-1}$, interpreted as <continuous> <class functions>. Its image is not the whole infinite-dimensional ring of <continuous> <class functions>; injectivity is the assertion being proved.
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