I'm reading a note having this definition: Let $R$ be a ring amd $M$ an $R$-module. The trivial extension of $R$ by $M$, written by $R\ltimes M$, is the $R$-algebra formed by endowing the direct sum $R\oplus M$ with the multiplication: $$(r,m)(s,n)=(rs,rn+sm).$$
Can you give me some references about this definition so that I can understand it better? This note also states some facts like if $R$ is Noetherian and $M$ is finitely generated then $R\ltimes M$ is Noetherian and has the same dimension as $R$. To prove these facts myself, I need to understand the ring structure on it, but I'm having some troubles understand its ideals and prime ideals.
A good reference is the survey "Idealization of a module" by Anderson and Winders, which you can find here (idealization is another name for trivial extension). The results you need are proved in Proposition 2.2 and Theorem 3.2.