An idempotent ring cannot be Morita equivalent to its ideal

TL;DR

The paper shows that an idempotent ring cannot be Morita equivalent to a proper idempotent ideal by leveraging Valjako’s enlargement framework and the notion of joint enlargements. It proves that if $R$ and an idempotent ideal $S$ of $R$ are Morita equivalent, then $R=S$, with a key step showing $R=RTR$ and $S=RSR$ within a joint enlargement. Consequently, standard unitalization constructions, such as the Dorroh extension and multiplier ring, cannot be Morita equivalent to the original nonunital ring. These results sharpen Morita theory for nonunital rings and clarify how enlargements constrain equivalence relations between rings and their ideals.

Abstract

In this note it is proven that an idempotent ring cannot be Morita equivalent to its idempotent proper ideal.

Theorems & Definitions (10)

• Definition 1.1: Def. 2.1 in Valjako
• Theorem 1.2: Theorem 3.9 in Valjako
• Lemma 2.1
• proof
• proof
• Theorem 2.3
• proof
• Corollary 3.2
• Corollary 3.4
• Remark 3.5