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For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every left $A$-module is a direct sum of simple objects, or equivalently that every object of $A$ is a finite direct sum of simple objects) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective and even free.)

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every left $A$-module is a direct sum of simple objects, or equivalently that every object of $A$ is a finite direct sum of simple objects) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective.)

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every object is semi-simple) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective and even free.)

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every left $A$-module is a direct sum of simple objects, or equivalently that every object of $A$ is a finite direct sum of simple objects) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective and even free.)

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every $A$-module is semi-simple) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective and even free.)

For an abelian category $A$, the following are equivalent:

1. Every short exact sequence splits.
2. Every object is projective.
3. Every object is injective.
4. Every additive functor from $A$ to an abelian category is exact.

(To show 4 $\Rightarrow$ 2, use the Hom functor.)

If $A$ has these properties, any full pseudo-abelian subcategory of $A$ is a Serre subcategory (in particular, abelian) with the same properties. They are also preserved by Serre localisation.

$A$ is semi-simple (in the sense that every object is semi-simple) provided every object of $A$ is Noetherian (or, equivalently, Artinian), or if it is a category of modules over an appropriate additive category. A fun counterexample is the category of infinite-dimensional vector spaces over a field, localised by the Serre subcategory of finite-dimensional vector spaces. I am not sure that it is Grothendieck (does it have infinite direct sums?) Another example, still in the spirit of Leonid's answer, is the category of finitely presented modules over a von Neumann regular ring $R$. (Reduce to the case of one generator to show projectivity. We get a quotient of $R$ by a finitely generated ideal. This ideal is generated by an idempotent, hence a direct summand, hence the quotient is indeed projective and even free.)