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

See also

• Divisible group
• Projective object