Difference between revisions of "User:Alec/Modules/Rings and modules"
(Created page with "==Translations== {| class="wikitable sortable" border="1" |- ! His ! My ! Comment |- ! Commutative ring | q-ring ({{AKA}}: cu-ring) | |- ! Ring | u-ring | |- !data...") |
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Latest revision as of 09:25, 16 October 2016
Translations
| His | My | Comment |
|---|---|---|
| Commutative ring | q-ring (AKA: cu-ring) | |
| Ring | u-ring | |
| [ilmath]\mathbb{Z}_n[/ilmath] | unsure | See: Integers modulo n |
Definitions
Other work
Possible distributivity page
{{Stub page|grade=A|msg=Proper stub page, I'd love a reference}} ==Definition== Let {{M|(R,\oplus,\odot)}} be a [[tuple]] consisting of a ''[[non-empty]]'' [[set]] {{M|R}}, and two [[binary operations]] (a kind of [[map]] where rather than writing {{M|f(a,b)}} we write {{M|afb}}): # {{M|\oplus:R\times R\rightarrow R}} # {{M|\odot:R\times R\rightarrow R}} Then we say for {{M|(R,\oplus,\odot)}} that: * "'''{{M|\oplus }} is left distributive over {{M|\odot}}'''" or "{{M|(R,\oplus,\odot)}} is left distributive" if - {{M|1=\forall a,b,c\in R[x\odot(y\oplus z)=(x\odot y)\oplus(x\odot z)]}} * "'''{{M|\oplus }} is right distributive over {{M|\odot}}'''" or "{{M|(R,\oplus,\odot)}} is right distributive" if - {{M|1=\forall a,b,c\in R[(x\oplus y)\odot z=(x\odot z)\oplus(y\odot z)]}} * "'''{{M|\oplus }} is distributive over {{M|\odot}}'''" or "{{M|(R,\oplus,\odot)}} is left distributive" if {{M|\oplus}} is both right and left distributive over {{M|\otimes}} Where it is understood that {{M|\oplus}} is the first operation in the tuple and {{M|\odot}} the second. Note that {{M|(R,\odot,\oplus)}} being left/right or just distributive is a totally different thing. There are slightly different forms of distributivity, but all of them have "first left-distributive over second if thing first (thing second thing) = (thing first
