A

topological ringis a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and $\cdot:R\times R\to R$ is a continuous map. Aleft Haar measureon $R$ is a (nonnegative) measure $\lambda$ on $R$ with respect to the Borel $\sigma$-algebra of $R$ such that there is a multiplicative map $l:R\to[0,\infty]$ called aleft multiplierfor which

- $l(x\cdot y)=l(x)\,l(y)$ for all $x,y\in R$,
- $\lambda(x\cdot S)=l(x)\,\lambda(S)$ for all $x\in R$ and for any Borel measurable subset $S\subseteq R$ (with respect to $\mathcal{T}$), and
- $\lambda$ is a Haar measure for the topological group $(R,+)$.
(We interpret $0\cdot \infty$ and $\infty\cdot 0$ as $0$.)

An example is the ring $\text{Mat}_{n\times n}(\mathbb{R})$ of $n$-by-$n$ matrices over $\mathbb{R}$ under the topology inherited from $\mathbb{R}^{n\times n}\cong\text{Mat}_{n\times n}(\mathbb{R})$ (here, $\cong$ means "is homeomorphic to"). If $\textbf{A}=\left[a_{i,j}\right]_{i,j=1,2,\ldots,n}$, then we take $\text{d}\lambda(\textbf{A})$ to be $\prod_{i=1}^n\,\prod_{j=1}^n\,\text{d}a_{i,j}$ (i.e., $\lambda$ is the Lebesgue measure of $\text{Mat}_{n\times n}(\mathbb{R})\cong \mathbb{R}^{n\times n}$). Then, $\lambda$ is a left Haar measure of $\text{Mat}_{n\times n}(\mathbb{R})$ with respect to the left multiplier $l(\textbf{X}):=\big|\det(\textbf{X})\big|^n$ for all $\textbf{X}\in\text{Mat}_{n\times n}(\mathbb{R})$.

Another example is $(\mathbb{C},+,\cdot)$. We take $\lambda$ to be such that $\text{d}\lambda(z):=\text{d}x\,\text{d}y$ for $z=x+\text{i}y$. Then, a left multiplier is $l(w):=|w|^2$ for each $w\in\mathbb{C}$. Similarly to the case of the ring of real $n$-by-$n$ matrices, $\text{Mat}_{n\times n}(\mathbb{C})$ has a left Haar measure $\lambda$ with respect to $l(\textbf{X}):=\big|\det(\textbf{X})\big|^{2n}$ for all $\textbf{X}\in\text{Mat}_{n\times n}(\mathbb{C})$, i.e., $\text{d}\lambda(\textbf{A}):=\prod_{i=1}^n\,\prod_{j=1}^n\,\left(\text{d}x_{i,j}\,\text{d}y_{i,j}\right)$ for $\textbf{A}=\left[x_{i,j}+\text{i}y_{i,j}\right]_{i,j=1,2,\ldots,n}$.

The third example here is $\mathbb{R}[T]/\left(T^n\,\mathbb{R}[T]\right)$ equipped with the topology inherited from $\mathbb{R}^n$. For $f(T)=\left(a_0+a_1T+\ldots+a_{n-1}T^{n-1}\right)+\left(T^n\,\mathbb{R}[T]\right)$, we take $\text{d}\lambda(f):=\prod_{i=0}^{n-1}\,\text{d}a_i$. Then, a left multiplier is $l(g):=\left|b_0\right|^n$, for $g(T)=\left(b_0+b_1T+\ldots+b_{n-1}T^{n-1}\right)+\left(T^n\,\mathbb{R}[T]\right)$.

Note that we can also similarly define the notion of

right Haar measuresfor rings and their correspondingright multipliers(which are usually denoted by $r$ in this thread).If a ring has both left and right Haar measures, then (due to commutativity of addition) the two Haar measures coincide (up to scalar multiple) and we can omit the adjectivesThe adjectives "left" and "right" only indicate which kind of multipliers the ring has. A ring with a Haar measure isleftandright, and simply call themHaar measures(ortwo-sided Haar measuresto be precise).unimodularif the left multiplier coincides with the right multiplier; otherwise, we call the ringnon-unimodular.

Then, $\text{Mat}_{n\times n}(\mathbb{R})$, $\mathbb{C}$ (as well as $\text{Mat}_{n\times n}(\mathbb{C})$), $\mathbb{R}[T]/\left(T^n\,\mathbb{R}[T]\right)$ (as well as $\mathbb{C}[T]/\left(T^n\,\mathbb{C}[T]\right)$), the ring $\mathbb{Z}_{p}$ of $p$-adic integers (see Crostul's comment below), and the ring of formal power series $\mathbb{F}_p[\![ T]\!]$ (see Jyrki Lahtonen's comment below) are examples of unimodular rings. Of course, many of these rings are commutative rings, and commutative rings with Haar measures are necessarily unimodular.

An example of non-unimodular rings is as follows. Consider $\mathbb{R}\times\mathbb{R}$ (some people may denote it by the *semidirect sum* $\mathbb{R}\,{\supset\!\!\!\!\!\!+}\,\mathbb{R}$) with the usual entry-wise addition, but with the twisted multiplication $(u,v)\cdot(x,y):=(ux,uy+v)$. Then, with the Haar measure $\text{d}\lambda(a,b):=\text{d}a\,\text{d}b$, we notice that a left multiplier of $\mathbb{R}^2$ is $l(u,v):=|u|^2$ for every $u,v\in\mathbb{R}$, whereas a right multiplier is $r(u,v):=|u|$ for every $u,v\in\mathbb{R}$. The ring $\text{Mat}_{n\times n}(\mathbb{R})\times\mathbb{R}^n$ (also denoted by $\text{Mat}_{n\times n}(\mathbb{R})\,{\supset\!\!\!\!\!\!+}\,\mathbb{R}^n$) and the ring $\text{Mat}_{n\times n}(\mathbb{C})\times\mathbb{C}^n$ (also denoted by $\text{Mat}_{n\times n}(\mathbb{C})\,{\supset\!\!\!\!\!\!+}\,\mathbb{C}^n$), with the usual entry-wise addition and with the twisted multiplication $(\mathbf{A},\mathbf{u})\cdot(\mathbf{B},\mathbf{v}):=(\mathbf{AB},\mathbf{Av}+\mathbf{u})$, are also non-unimodular for the same reason.

Has there been any study on this sort of concepts? I'm sure that I am not the first who thought about the notion of

Haar measuresfor rings. Are there any criteria for a ring to be unimodular? Are there rings with Haar measures of a given characteristic $k>0$ which are non-unimodular, or simple, or at least non-commutative? Are there rings with left Haar measures, but without right Haar measures? (Unlike groups, I expect to find rings with only one kind of Haar measures.) The cardinality of each example of rings with Haar measures I have so far is that of the continuum $\mathfrak{c}=2^{\aleph_0}$. Do there exist rings with Haar measures of higher cardinalities? Please let me know if you have or know of any references.On the other hand, there may be some pathology with this concept of Haar measures for rings, and because of that, nobody introduces this concept. If that is the case, would you please let me know why?

EDIT I: I removed a wrong example from the question. It turns out that, with respect to the discrete topology, $(\mathbb{Z},+,\cdot)$ does not have a Haar measure.

EDIT II: I think I may need another compatibility condition, as stated in menag's reply below. That is, I may need to enforce the following condition (which I shall name it *left compatibility*): for any Borel measurable subset $S\subseteq R$ and for every $x\in R$, $x\cdot S$ is also Borel measurable. For right Haar measures, we may similarly define *right compatibility*. Alternatively, I may need to modify Condition 2 to $\lambda(x\cdot S)=l(x)\,\lambda(S)$ for each Borel measurable subset $S\subseteq R$ and for any $x\in R$ such that $x\cdot S$ is Borel measurable (and likewise for right Haar measures). I am in favor of the former modification (i.e., the left or right compatibility condition) and shall assume it in all instances.

EDIT III: I may be wrong about this claim: "An abelian group has at most one Haar measure (up to scalar multiple)." (See the **boldfaced** portion of my question.) Without the locally compact Hausdorff assumption, there may be an abelian group with two essentially different Haar measures. If the underlying additive group of a ring is isomorphic to such an abelian group, then the ring may have a left Haar measure and a right Haar measure which are not proportional. Maybe somebody who knows better about topological groups can give me some insight.