Ion. D. Ion and Constantin Nita:Ultraproducts of transitive rings of linear transformations, p.55-60

Abstract:

If $F$ is a nonprincipal ultrafilter on an infinite set $I$, then for any family $\left\{{M_i}\right\}_{i\in I}$ of modules ${M_i}\in{R_i}$-mod we have a natural immersion $\varphi:\left(\displaystyle \prod_{i\in I}End_{R_i}\left(M_i\right)\right)_F$ $\rightarrow$ $End_{R_F}\left(M_F\right)$ where $R_F=\left(\displaystyle \prod_{i\in I}R_i\right)_F$ and $M_F=\left(\displaystyle \prod_{i\in I}M_i\right)_F$ (theorem 1.1). Generally, $\varphi$ is not an isomorphism as we can see in Examples 1.2 and 1.3. An ultraproduct of $2-$transitive rings of linear transformations is a $2-$transitive ring (theorem 2.2). As a consequence we obtain a classical result which says that the immersion $\varphi$ in theorem 1.1 is an isomorphism in case each $M_i$ is a simple faithful module (corollary 2.5). Finally we prove a result with applications in PI-theory: an ultraproduct of closed primitive rings is a closed primitive ring.

Key Words: Ultraproducts, rings of linear transformations, m-transitive rings, closed primitive rings.

2000 Mathematics Subject Classification: Primary: 03C20,
Secondary: 03C60, 16S50, 15A04, 16D60.

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