Difference between revisions of "Direct sum (ring)"
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Latest revision as of 14:32, 8 June 2015
For other kinds of direct sums see Direct sum
Contents
[hide]Definition
Given two rings (R,+R,×R) and (S,+S,×S) their direct sum is defined on the set R×S (where × is the Cartesian product), that is:
- R×S={(x,y)| x∈R∧y∈S}
and is denoted:[1]
- R⊕S=(R×S,+,×) or simply R⊕S as Mathematicians are lazy
where the operation + and × are defined as follows:
- Given (x,y), (x′,y′)∈R⊕S we define:
- Addition as: (x,y)+(x′,y′)=(x+x′,y+y′) or more formally (x,y)+(x′,y′)=(x+Rx′,y+Sy′)
- Multiplication as: (x,y)(x′,y)=(xx′,yy′) or more formally (x,y)(x′,y′)=(x×Rx′,y×Sy′)
Other group properties
Unity
Commutative
[Expand]
Theorem: R⊕S is a commutative ring if and only if both R and S are commutative rings
