Example of subdirect product of the family of near-ring $S_i$

Question

I am studying a near-ring theory by $\textbf{Gunter Pilz}$. I want some example of near-ring which is subdirect product of the family of near-ring $S_i$. Secondly i have doubt in the definition of subdirectly irreducible.

• A near-ring $R$ is a subdirect product of a family of near-rings $\{S_i \;:\; i ∈\in I\}$ if there is a monomorphism $k: R \rightarrow S = \underset{i \in I}{\prod} S_i $ such that $\pi_i o k$ is an epimorphism for all $i \in I$, where $\pi_i: S \rightarrow S_i$ is the canonical epimorphism.

• A near-ring $R$ is called subdirectly irreducible if $R$ is not isomorphic to a non-trivial subdirect product of near-rings.

Any help would be appreciated. Thank you.

Answer 1

Let $F$ be a field.

$F$ is a subdirect product with one factor. Too trivial? Then ...

$F\times F$ is a subdirect product. Not impressed?

$\bigoplus_{i=1}^\infty F$ is a nearring without identity, and you can call this ring $R$ or adjoin an identity and call it $R$. Either way $R$ is a subdirect product $R\hookrightarrow \prod_{P\in \mathcal C}R/P$ where $\mathcal C$ is the collection of prime ideals of $R$.

I have no idea what your doubt question is about the definition of subdirectly irreducible nearrings is, since you did not mention it (you just wrote a definition.) The normal definition of a subdirectly irreducible object is that whenever $\phi:R\hookrightarrow \prod_{i\in I}R_i$ is a subdirect product, then $\pi_i\phi$ is an isomorphism for one of the $i$'s.