module homomorphism

module homomorphism

Let

R

be a ring and

M

,

N

left modules

over

R

. A function

f

:

M

→

N

is said to be a

left module

homomorphism

(over

R

) if

1.

f

is additive:

f

⁢

(

u

+

v

)

=

f

⁢

(

u

)

+

f

⁢

(

v

)

, and

2.

f

preserves left

scalar multiplication

:

f

⁢

(

r

⁢

v

)

=

r

⁢

f

⁢

(

v

)

, for any

r

∈

R

.

If

M

,

N

are right

R

-modules, then

f

:

M

→

N

is a

right module homomorphism

provided that

f

is additive and preserves right scalar multiplication:

f

⁢

(

v

⁢

r

)

=

f

⁢

(

v

)

⁢

r

for any

r

∈

R

. If

R

is

commutative

, any left module homomorphism

f

is a right module homomorphism, and vice versa, and we simply call

f

a

module homomorphism

.

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.

Title

module homomorphism

Canonical name

ModuleHomomorphism

Date of creation

2013-03-22 19:22:54

Last modified on

2013-03-22 19:22:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D20

Classification

msc 15-00

Classification

msc 13C10

Classification

msc 16D10

Defines

bimodule homomorphism

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