I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define multiplication by
\begin{equation*}
(a,m)(b,n) := (ab,\,an+mb)~.
\end{equation*}
The set $A\times N$ equipped with this multiplication is a (unital associative) ring with the
two-sided ideal $0\times N$ whose product with itself is $0$. The ideal $0\times N$ is the kernel of the surjective homomorphism of rings $A\times N\to A : (a,n)\mapsto a$. If $A$ is Jacobson-semisimple,
then $0\times N$ is the Jacobson radical of the ring $A\times N$.
Does this construction of the ring $A\times N$ from a ring $A$ and an $A$-$A$ bimodule $N$ have a name, and perhaps an established notation? The notation $A\times N$ is misleading since it suggests a direct product of rings, which it is not.
[Added a day later.]
The comment by Dag Oskar Madsen to the answer by Jeremy Rickard got me thinking.
First consider the definition of the inner semidirect product of groups.
Let $G$ be a group with a subgroup $H$ and a normal subgroup $N$. The following statements are equivalent:
$G=NH$ ($=HN$) and $N\cap H=\{e\}$.
The natural embedding $H\to G$, composed with the natural projection $G\to G/N$,
is an isomorphism of groups.There exist a group $H'$ and homomorphisms of groups $p\colon G\to H'$ and $j\colon H'\to G$
such that $p\circ j = \mathrm{id}_{H'}$ and $j(H')=H$, $\ker p = N$.There exists an idempotent endomorphism $e$ of the group $G$ such that $e(G)=H$
and $\ker e = N$.
If one of these statements holds (and therefore all hold) we say that the group $G$ is the (inner) semidirect product of its normal subroup $N$ and its subgroup $H$, and write $G=N\rtimes H$.
Now compare this definition to an analogous situation with a ring in place of a group.
Let $R$ be a ring with a subring $A$ and a (two-sided) ideal $N$. The following statements are equivalent:
The underlying additive group of $R$ is a direct sum of the additive underlying groups
of $N$ and $A$, which we write $R=N\oplus A$.The natural embedding $A\to R$, composed with the natural projection $R\to R/N$,
is an isomorphism of rings.There exist a ring $A'$ and homomorphisms of rings $p\colon R\to A'$ and $j\colon A'\to R$
such that $p\circ j=\mathrm{id}_{A'}$ and $j(A')=A$, $\ker p = N$.There exists an idempotent endomorphism $e$ of the ring $R$ such that $e(R)=A$
and $\ker e=N$.
I propose to say, whenever the situation described by any of the four cases above occurs,
that $R$ is the (inner) semidirect product of its ideal $N$ and its subring $A$,
and write $R=N\rtimes A$.
The structure of the inner semidirect product of an ideal and a subring of a ring
suggests the following definition of the outer semidirect product
of a rng $N$ and a ring $A$ with respect to
a coherent biaction $\varphi$ of $A$ on $N$.
(A rng is an additive group with an associative biadditive multiplication.
A multiplicative identity is not required; if it is present, it is ignored.)
What we mean by a coherent biaction of $A$ on $N$:
the rng $N$ is an $A$-$A$ (unital) bimodule,
where $(am)n=a(mn)$, $m(na)=(mn)a$ for all $m$, $n$ in $N$ and all $a$ in $A$.
We define the multiplication on the set $B:=N\times A$ by
\begin{equation*}
(m,a)(n,b) \,:=\, (mn+an+mb,\,ab)~.
\end{equation*}
Then $B$ is a ring, $0\times A$ is a subring of $B$, $N\times 0$ is an ideal of $B$,
and $B=(N\times 0)\oplus(0\times A)$.
I propose to call the ring $B$ the (outer) semidirect product
of a rng $N$ and a ring $A$ with respect to
a coherent biaction $\varphi$ of $A$ on $N$,
and write $B = N\rtimes_\varphi A$.
I believe this is a natural transfer, by analogy, of the notion of a semidirect product
from groups to rings.
The outer semidirect product for rings is peculiar in that it constructs a ring
from a rng and
a ring
(thus it is an 'inter-species' construction)
with the ring coherently biacting on the rng.
If we have just a plain $A$-$A$ bimodule $N$,
with no multiplication on $N$,
we equip $N$ with the
all-zero multiplication
$($$mn=0$ for all $m$, $n$ in $N\,$$)$, obtaining a legitimate rng,
and so make $A$ coherently biacting on the rng $N$.
I propose that in this special case we write $B=N\mathbin{{}_0\rtimes_\varphi} A$,
where $0$ stands for the all-zero multiplication on $N$
and $\varphi$ is the bimodule action of the ring $A$
on the additive group $N$.
I googled "semidirect product of rings" and got no exact matches.
The approximate matches
were "crossed product of rings", "semidirect product of Hopf algebras",
"semidirect product of Lie algebras/rings".
There was also a 'mixed marriage' semidirect product $K[N]\rtimes_\varphi H$
$($isomorphic
to the group algebra $K[N\rtimes_\varphi H]$$)$
of the group algebra of a group $N$ with coefficients in
a commutative ring $K$, and a group $H$,
with respect to an action $\varphi$ of the group $H$ on the group $N$
(by isomorphisms),
which induces an action of the group $H$ on the group algebra $K[N]$
(by isomorphisms).
Mark the notion of a retract of a topological space:
a subspace $A$ of a topological space $X$
is a retract of $X$
if there exists an idempotent continuous map $e\colon X\to X$ such that $e(X)=A$;
the restriction of $e$ to $r\colon X\to A$
(the codomain $X$ of $e$ is narrowed down to the codomain $A$ of $r$)
is a retraction.
