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Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Is the endomorphism ring of a uniserial object local?

0. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Is the endomorphism ring of a uniserial object local?

0. Assume $A$ has the property that every indecomposable object is uniserial. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<...<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Is the endomorphism ring of a uniserial object local?

0. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Is the endomorphism ring of a uniserial object local?

0. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<...<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<...<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

0. Is the endomorphism ring of a uniserial object local?

0. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

1. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?