7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings
Polynomial Rings
Let $R$ a commutative unital ring. We define $R[x]$ to be formal sums in the indeterminant $x$ and its powers of the form
\[R[x] = \left\{\sum_{i=0}^n a_i x^i \mid a_i \in R, n\in\mathbb{Z}_{\ge 0}\right\}.\]The above ring, along with the obvious choice of addition and multiplication, is called the polynomial ring in $x$ with coefficients in $R$. Suppose $p(x) = a_n x^n + \cdots + a_1 x + a_0 \in R[x]$. If $a_n \neq 0$, we say that $p(x)$ has degree $n$, $a_n x^n$ is called the leading term, and $a_n$ is the leading coefficient. If the leading coefficient is $1$, then the polynomial is monic.
Proposition. Let $R$ be an integral domain and let $p(x), q(x) \in R[x]$ be nonzero. Then $\deg(p(x)q(x)) = \deg p(x) + \deg q(x)$
Proof. If not, then the leading coefficient is zero and so the leading coefficients of $p(x)$ and $q(x)$ form a pair of zero divisors in $R$, contradiction.
Notice that if $R$ has zero divisors, then $R[x]$ will as well since there is a copy of $R$ embedded in $R[x]$.
Matrix Rings
Let $R$ be a ring and $n$ be a positive integer. Then the set $M_n(R)$ of $n\times n$ matrices with entries from $R$ forms a ring under entrywise addition and matrix multiplication.
Proposition. If $R$ is any nontrivial ring and $n \ge 2$, then $M_n(R)$ is a noncommutative ring.
Proof. Take $a, b \in R$ such that $ab \neq 0$. Let
\[A = \begin{bmatrix} a & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix} \quad\text{ and }\quad B = \begin{bmatrix} 0 & b & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix}.\]Then $AB$ is the matrix with $ab$ as the $(1,2)$-entry and $0$ everywhere else. $BA$, on the other hand, is the zero matrix.
Remark. Notice that the product $BA$ shows that every $M_n(R)$ has zero divisors for nontrivial $R$ when $n \ge 2$.
Definition. We say that $A$ is a scalar matrix if all of its main diagonal entries are equal to some fixed $a \in R$.
Here are a couple of immediate results that we will state without proof.
Proposition. The set of all such matrices forms a subring of $M_n(R)$ isomorphic to $R$.
Proposition. The scalar matrices commute with all elements of $M_r(R)$.
Definition. Let $R$ be a unital ring. Then the subring formed by the collection of invertible $n\times n$ matrices in $M_n(R)$ is called the general linear group $\mathrm{GL}_n(R)$ of degree $n$ over $R$.
One more immediate result.
Proposition. If $S$ is a subring of $R$, then $M_n(S)$ is a subring of $M_n(R)$.
Group Rings
Definition. Let $R$ be a nontrivial commutative unital ring and let $G = {g_1, \ldots, g_n}$ be a finite group with group operation written multiplicatively. The group ring $RG$ of $G$ with coefficients in $R$ is the set of all formal sums
\[a_1g_1 + a_2g_2 + \cdots + a_n g_n, \qquad a_i \in R, \quad 1 \le i \le n.\]Addition is defined componentwise and multiplication is defined extending $(ag_i)(bg_j) := (ab)(g_i g_j)$ in the obvious way. Note that $RG$ is commutative if and only if $G$ is abelian.
| Proposition. If $ | G | > 1$, then $RG$ has zero divisors. |
Proof. Let $g$ be a non-identity element. Then
\[(1 - g)(1 + g + \cdots + g^{|g|-1}) = 0.\]Proposition. If $S$ is a subring of $R$, then $SG$ is a subring of $RG$.
Exercises
Problem 3. Define the set $R[[x]]$ of formal power series in the indeterminate $x$ with coefficients from $R$ to be all formal infinite sums
\[\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2x^2 + a_3x^3 + \cdots.\]Define addition and multiplication as for series with real or complex coefficients.
- (a) Prove that $R[[x]]$ is a commutative ring with $1$.
- (b) Show that $1 - x$ is a unit in $R[[x]]$ with inverse $1 + x + x^2 + \cdots$.
- (c) Prove that $\sum_{n=0}^\infty a_n$ is a unit in $R[[x]]$ if and only if $a_0$ is a unit in $R$.
Proof. (a) Clearly, $1$ is the $1$.
(b) Multiply the two in both orders. Both end up being $1$.
(c) For the forward direction, let $\sum_{n=0}^\infty b_n$ be the multiplicative inverse and multiply the two. The constant term necessitates that $a_0$ is a unit. For the converse direction, we can build up the multiplicative inverse of the power series recursively to give a concrete existence proof or we can proceed by contrapositive and show that the constant term of the product of any power series against the original one can not be inverted since $a_0$ is not a unit.
Problem 4. Prove that if $R$ is an integral domain then the ring of formal power series $R[[x]]$ is also an integral domain.
Proof. Pick two nonzero elements in $R[[x]]$ that are nonzero. Then the coefficient of the lowest degree term in their product is given by the product of the coefficients of the lowest degree terms in each respective factor. Since $R$ is an integral domain, this coefficient product cannot be zero so the power series is also not zero.