Zero module

In mathematics , the zero module is a module that consists of only one element, the zero element . In the category of modules above a given ring , the null module is the null object .

definition

The zero module is a module over any ring consisting of the single element set provided with the only possible addition given by ${\displaystyle (\{0\},+,\cdot )}$ ${\displaystyle R}$ ${\displaystyle \{0\}}$

${\displaystyle 0+0=0}$

and the only possible multiplication given by

${\displaystyle r\cdot 0=0}$

for all elements . ${\displaystyle r\in R}$

Category theory

In the category of all modules over a given ring with the module homomorphisms as morphisms , the zero module is the zero object : From each module there is exactly one homomorphism into the zero module and from the zero module there is exactly one homomorphism in each module, namely the null function that corresponds to the respective null morphism .

See also

• Zero vector space

literature

• Michael Artin : Algebra. Birkhäuser, Basel et al. 1998, ISBN 3-7643-5938-2 .