# Homomorphism from a simple module to a direct sum of itself

Let $$R$$ be a ring and $$T$$ be a simple left $$R$$-module. Consider a nonzero $$R$$-linear map $$h:T\to\bigoplus_{i\in I}{T_i},$$ where $$(T_i)_{i\in I}$$ is a nonempty family of $$R$$-modules isomorphic to $$T$$. For convenience, we may identify each $$T_i$$ with its canonical copy in $$\bigoplus_{i\in I}{T_i}$$.

Conjecture. $$h(T)=T_i$$ for exactly one $$T_i$$.

Here are my attempts: Because $$h$$ is nonzero and $$T$$ is simple, by Schur's lemma, the map $$h$$ is injective, namely $$h(T)\cong T$$.

Besides, since $$h$$ is nonzero, we can find one $$i\in I$$ such that $$h(T)\cap T_i\ne 0$$. However, by the simplicity of $$h(T)$$ and $$T_i$$, we must have $$h(T)=T_i$$ in this case.

Then according to the property of direct sum, for $$j\ne i$$, $$h(T)\cap T_j=T_i\cap T_j=0,$$ proving the uniqueness.

Edit. My arguments above are false. Consider $$T=\operatorname{Span}([1,1])\subseteq\mathbb{R}^2$$, where $$T_1=\operatorname{Span}([1,0])$$ and $$T_2=\operatorname{Span}([0,1])$$ in which $$h:T\to T_1\oplus T_2=\mathbb{R}^2$$ is the inclusion map. But $$h(T)\cap T_1=h(T)\cap T_2=0$$ in this case.

Nevertheless, I still wonder if it is possible to show that

New Conjecture. $$h(T)\subseteq\bigoplus_{i\in J}{T_i}$$ for some finite subset $$J\subseteq I$$.

Any help will be appreciated.

Every simply module is generated by one element. Let's call that element $$u$$. Then $$h(u)$$ is an element of a direct sum, hence co-finitely many of its 'coordinates' are zero. Letting $$J$$ be the set of all $$i$$ for which the $$i$$th coordinate of $$h(u)$$ is nonzero provides an affirmative answer to your question.
• I'll be honest --- I thought it was false. You could take countably many copies of Z, say, and send $u$ to $(1,1,1,...)$. But then I realized you'd said "simple module", and I had to look up the definition. And somewhere in Wikipedia it pointed out that simple modules are cyclic. And I had to look up direct sum to see whether it was the one with the "cofinite" restriction (vs direct product) -- I can never remember that(I haven't been a real mathematician for 30+ years now). And with those two lookups, the rest just fell out. So answering your question taught me a good deal as well. Dec 29, 2023 at 4:12