# Injective object

Injective object is a term from the mathematical branch of category theory .

## definition

In a category an object is injective if for any monomorphism and every one is, so is. ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle Q}$ ${\ displaystyle \ alpha \ colon A \ rightarrow B}$ ${\ displaystyle f \ colon A \ rightarrow Q}$ ${\ displaystyle f ^ {*} \ colon B \ rightarrow Q}$ ${\ displaystyle f ^ {*} \ circ \ alpha = f}$ Accordingly, it is injective if and only if the induced mapping is surjective for all monomorphisms . ${\ displaystyle Q}$ ${\ displaystyle \ alpha \ colon A \ rightarrow B}$ ${\ displaystyle \ mathrm {Mor} _ {\ mathcal {C}} (B, Q) \ ni g \ mapsto g \ circ \ alpha \ in \ mathrm {Mor} _ {\ mathcal {C}} (A, Q )}$ ## Examples

• In the category of sets Me , every set is injective.
• Injective objects in the category of Abelian groups are the divisible groups ; H. those groups for which multiplication by a non-zero integer is surjective; Examples are and .${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Q} / \ mathbb {Z}}$ • In the category of vector spaces over a body, every object is injective.
• Every terminal object in a category is injective.
• If there is a family of injective objects, the product of that family is injective if it exists.${\ displaystyle (Q_ {i} | i \ in I)}$ • If the category has a null object, then a product of injective objects is injective if and only if each is injective.${\ displaystyle Q_ {i}}$ • If injective, then every monomorphism is a cut (that is, there is a with ).${\ displaystyle Q}$ ${\ displaystyle \ alpha \ colon Q \ rightarrow P}$ ${\ displaystyle \ beta \ colon P \ rightarrow Q}$ ${\ displaystyle \ beta \ circ \ alpha = \ mathbf {1} _ {Q}}$ • In the category of topological spaces, the set is not injective because the inclusion map is not a section. There is no such thing as a continuous surjective function . This is a consequence of the intermediate value theorem .${\ displaystyle \ {- 1,1 \}}$ ${\ displaystyle \ {- 1,1 \} \ rightarrow \ mathbb {R}}$ ${\ displaystyle \ beta \ colon \ mathbb {R} \ rightarrow \ {- 1,1 \}}$ ## Injective modules

For a right module over a ring , the following statements are equivalent. ${\ displaystyle Q}$ ${\ displaystyle R}$ 1. ${\ displaystyle Q}$ is injective in the category of legal modules.
2. For every monomorphism there is one with . It is the identity .${\ displaystyle \ alpha \ colon Q \ rightarrow M}$ ${\ displaystyle \ beta \ colon M \ rightarrow Q}$ ${\ displaystyle \ beta \ circ \ alpha = \ mathbf {1} _ {Q}}$ ${\ displaystyle \ mathbf {1} _ {Q}}$ ${\ displaystyle Q}$ 3. Baer's criterion : For every legal ideal and every homomorphism there is one such that is.${\ displaystyle {\ mathfrak {a}} \ hookrightarrow R}$ ${\ displaystyle f \ colon {\ mathfrak {a}} \ rightarrow Q}$ ${\ displaystyle f ^ {*} \ colon R \ rightarrow Q}$ ${\ displaystyle f ^ {*} \ circ \ iota = f}$ Injective modules were introduced in 1940 by Reinhold Baer who, however, used the adjective complete instead of injective . The German designation injective module can be proven in 1953.

### Examples

1. A ring is semi-simple if and only if every module above the ring is injective. Hence every vector space over a sloping body is injective. From Baer's criterion it follows that precisely the divisible modules are injective via main ideal rings. A module is divisible if and only if is for all ring elements .${\ displaystyle Q \ cdot r = Q}$ ${\ displaystyle r}$ 2. If there is a family of modules, then the direct product of the family is injective if and only if each is injective.${\ displaystyle (Q_ {i} | i \ in I)}$ ${\ displaystyle Q_ {i}}$ 3. A ring is Noetherian if and only if the direct sum of injective modules is injective. This is a generalization of the corresponding statement about divisible Abelian groups.
4. Above a hereditary (hereditary) ring, each epimorphic image of an injective module is injective. This is a generalization of the corresponding theorem about divisible groups.
5. A torsion-free module is injective via an integrity ring exactly when it can be divided.
6. If there is a unitary ring homomorphism, there is an S-module on both sides. If there is another S-module, then the set of S-homomorphisms on the right-hand side carries through an R-module structure . The following applies: If an S module is injective, then an injective R module is. This is especially important in the case . If a divisible group is injective as a module, it is an injective R module.${\ displaystyle \ rho \ colon S \ rightarrow R}$ ${\ displaystyle R}$ ${\ displaystyle Q}$ ${\ displaystyle \ operatorname {Hom} _ {S} (R, Q)}$ ${\ displaystyle (\ alpha \ cdot r) (x) \ colon = \ alpha (r \ cdot x)}$ ${\ displaystyle Q}$ ${\ displaystyle \ operatorname {Hom} _ {S} (R, Q)}$ ${\ displaystyle S = \ mathbb {Z}}$ ${\ displaystyle D}$ ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ operatorname {Hom} _ {\ mathbb {Z}} (R, D)}$ ### There are enough injective modules

Each module can be mapped monomorphically into an injective module. ${\ displaystyle M}$ ### Injective sheath

A sub-module is called large if it is the only sub-module of that has an average . A monomorphism is called essential if large is in. The following applies: ${\ displaystyle U \ hookrightarrow Q}$ ${\ displaystyle \ {0 \}}$ ${\ displaystyle Q}$ ${\ displaystyle U}$ ${\ displaystyle \ {0 \}}$ ${\ displaystyle \ alpha: M \ rightarrow Q}$ ${\ displaystyle \ alpha (M)}$ ${\ displaystyle Q}$ Each module can essentially be mapped into an injective module . The module is uniquely determined by this property except for isomorphism. It is called the injective hull of M and is often referred to as. ${\ displaystyle Q}$ ${\ displaystyle Q}$ ${\ displaystyle I (M)}$ ## Indecomposable injective modules

A module is called directly indivisible if it is not the direct sum of two sub- modules not equal to zero. The following statements are equivalent for a module . ${\ displaystyle M}$ ${\ displaystyle M}$ 1. Any sub-module other than the null module is large in .${\ displaystyle M}$ 2. The injective shell cannot be dismantled directly.${\ displaystyle I (M)}$ 3. ${\ displaystyle I (M)}$ is the injective envelope of each sub-module not equal to zero.
4. The ring of endomorphisms of is local .${\ displaystyle I (M)}$ A module that satisfies the equivalent properties of the set is called uniform . is then often called irreducible (average irreducible ). ${\ displaystyle M}$ ### Examples

• Every simple module is uniform, so it has an injective shell that cannot be dismantled.
• If there is a prime ideal in the commutative ring , then is uniform. In particular, every integrity ring is uniform as a module.${\ displaystyle {\ mathfrak {p}} \ hookrightarrow R}$ ${\ displaystyle R}$ ${\ displaystyle R / {\ mathfrak {p}}}$ ## Individual evidence

1. ^ Friedrich Kasch, "Modules and Rings", Teubner, Stuttgart 1977, page 113, ISBN 3-519-02211-7
2. Reinhold Baer : Abelian groups that are direct summands of every containing abelian group . In: Bulletin of the American Mathematical Society . tape 46 , no. October 10 , 1940, p. 800-806 , doi : 10.1090 / S0002-9904-1940-07306-9 .
3. B. Eckmann , A. Schopf: About injective modules . In: Archives of Mathematics . tape 4 , no. 2 , April 1953, p. 75-78 , doi : 10.1007 / BF01899665 .
4. ^ Friedrich Kasch, "Modules and Rings", Teubner, Stuttgart 1977, page 114, ISBN 3-519-02211-7
5. ^ Friedrich Kasch, "Modules and Rings", Teubner, Stuttgart 1977, page 118, ISBN 3-519-02211-7