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mathStudent
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Question on simple modules and projective covers

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This ishave the proof of Lemma 2.2

enter image description herefollowing question:

$A$ is assumed to be an Artin algebra and mod(A) the category of finitely generated $A$ - modules. By $S \rightarrow I_s$ one means the injective hull of a simple module $S$. $P_s \rightarrow S$ denoted the projective cover of $S$.

Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ be module homomorphism with $f \neq 0$. Then $S_1 \cong S_2$.

The sentence underlined in pink confuses me the most. How does it follow form $g \neq 0$ that $S_r=S$? Any help is highly appreciated!

Question on simple modules

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is the proof of Lemma 2.2

enter image description here

$A$ is assumed to be an Artin algebra and mod(A) the category of finitely generated $A$ - modules. By $S \rightarrow I_s$ one means the injective hull of a simple module $S$. $P_s \rightarrow S$ denoted the projective cover of $S$.

The sentence underlined in pink confuses me the most. How does it follow form $g \neq 0$ that $S_r=S$? Any help is highly appreciated!

Question on simple modules and projective covers

I have the following question:

Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ be module homomorphism with $f \neq 0$. Then $S_1 \cong S_2$.

Any help is highly appreciated!

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mathStudent
mathStudent
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On the radical of a module category Question on simple modules

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mathStudent
mathStudent
  • 251
  • 1
  • 6

On the radical of a module category

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is the proof of Lemma 2.2

enter image description here

$A$ is assumed to be an Artin algebra and mod(A) the category of finitely generated $A$ - modules. By $S \rightarrow I_s$ one means the injective hull of a simple module $S$. $P_s \rightarrow S$ denoted the projective cover of $S$.

The sentence underlined in pink confuses me the most. How does it follow form $g \neq 0$ that $S_r=S$? Any help is highly appreciated!