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partially ordered ring
A ring $R$ that is a poset at the same time is called a partially ordered ring, or a poring, if, for $a,b,c\in R$,

$a\leq b$ implies $a+c\leq b+c$, and

$0\leq a$ and $0\leq b$ implies $0\leq ab$.
Note that $R$ does not have to be associative.
If the underlying poset of a poring $R$ is in fact a lattice, then $R$ is called a latticeordered ring, or an lring for short.
Remark. The underlying abelian group of a poring (with addition being the binary operation) is a pogroup. The same is true for lrings.
Below are some examples of porings:

Clearly, any (totally) ordered ring is a poring.

The ring of continuous functions over a topological space is an lring.

Any matrix ring over an ordered field is an lring if we define $(a_{{ij}})\leq(b_{{ij}})$ whenever $a_{{ij}}\leq b_{{ij}}$ for all $i,j$.
Remark. Let $R$ be a poring. The set $R^{+}:=\{r\in R\mid 0\leq r\}$ is called the positive cone of $R$.
References
 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
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Comments
Additional more commonly known definition
Please add this more commonly known definition of a poring (I wrote it down in the preliminary chapter of my dissertation) more known in the field of real algebraic geometry where most rings dealt with are commutative and unitary:
Given a commutative unitary ring $A$, a subset $A^+$ of $A$ is called a \emph{partial ordering} if it has the following property.
\begin{itemize}
\item $a\in A \Rightarrow a^2 \in A$
\item $x,y\in A^+ \Rightarrow x+y,xy \in A^+$
\item $x,x \in A^+ \Leftrightarrow x=0$
\end{itemize}
We say that a ring \emph{$A$ has a partial ordering $A^+$}, if there exist a subset $A^+\subset A$ such that $A^+$ is a partial ordering of $A$. We usually write it as a pair $(A,A^+)$ and call this a \emph{partially ordered ring} or a \emph{poring}
Remark:
If a commutative unitary ring $A$ has a partial ordering $A^+$ then set
$$\{a^2 : a\in A\} $$
is also a partial ordering of $A$. This partial ordering is, for obvious reasons, called the \emph{weakest partial ordering of $A$}. It is also known as the \emph{sum of squares of $A$}.
Re: Additional more commonly known definition
The idea I gave on defining Partial ordering can also be defined for noncommutative ring. So I don't see why we would not do so
In your remark you defined a "positive cone", which is indeed another name for a "partial ordering" (maybe positive cone is a better name, as partial ordering are words coming from concept of "orders").
Another remark:
There is a bijection between partial orderings (positive cones) and partial order (as you original defined)
this is given by:
$P \mapsto <_P $
where P is a partial ordering, and $<_P$ is the partial order defined by
$x<_P y$ iff $yx \in P$.
In your first remark you just gave the inverse of this bijection I defined.
jocaps
Birkhoff
I wonder, does Birkhoff specifically define lrings? I never read Birkhoff so I thought I give it a look. Well.. I know some people could swear by this book (I know at least 2 people who have "enjoyed it"), but after taking one look at it.. I think my eyes hurt.. I don't know maybe the way he writes isn't suitable to me.
But anyway, back to my question. Could you maybe tell me the page where Birkhoff specifically defines lrings? I know the definition of lrings to be the same as yours except that its without the first condition for your porings. i.e. for me lrings are. (I learned them from a paper from Melvin Henriksen and J.R. Isbell.)
rings with partial order (in the sense we know it) that makes them a lattice and such that the following condition holds:
x>=0, y>=0 => xy >= 0 (so the first condition is not there).
Maybe the first condition follows? Henriksen and Isbell refer a lot to the work of Birkhoff regarding works on lrings .. so i do suppose they have the same notion of lrings as Birkhoff. So I am not sure, either I am trying to see if the first condition of poring holds for lrings or if your definition of lring is exactly what Birkhoff stated in his book "Lattice Theory".
Re: Birkhoff
Unfortunately, I don't have the book with me.. I read it in the library. It's a classic for people who are into lattices. He's really the guy that brought lattices onto the same footing as other famous algebraic concepts such as groups and rings.
I recall it's towards the end of the book... Take a nap or use some eye drops, and read the book again.. It will certainly be faster than me driving to the library to find the book, verify it, and tell you afterwards.