Understanding Rings: Defining Addition and Multiplication in Abstract Algebra

[IB1]

I need to learn some abstract algebra, and it's pretty hard doing this on my own. Please help me.

According to the definition, Ring is an algebraic structure with two binary operations , commonly called addition (+) and multiplication ( . ). We write (R,+, .). Some examples of rings are: (Z, +, .), (2Z, +, .), (Q, +, .)etc...My problem is that where it is written (R,+,.), is + and . normal addition and multiplication respectively, or they're just some binary operations? If they were some binary operations, why they're saying that (Z, +, .) is a ring, without defining + and . ??

Sorry if my question were silly.

 

[Mark44]

You can assume that + and . represent the usual operations for addition and multiplication, unless they are defined differently in the problem. (Z, +, .) is the ring of integers, with the usual operations for addition and multiplication.

 

[IB1]

Yes, but why then in the begging of the chapter we develop arithmetic on rings if + and . are simply addition and multiplication? Moreover, the third axiom [ c(a+b)=ca +cb, (a+b)c=ac+bc] would be trivial and we wouldn't need to check for it ever because it is always true.

Sorry for bombing with questions :blush:

 

[Mark44]

The distributivity axioms are true for the usual addition and multiplication operations on integers, but for some sets and some operations, they don't hold. The beginning of the chapter is showing you how to verify the axioms on some simple rings.

 

[IB1]

Well, I can understand that distributivity might not hold for "some sets and some operations" ... but still it's written + and . and so the operations are addition and multiplication ... I could understand if it were written like this [c#(a*b)=c#a*c#b] ?

So, Mark44, are you suggesting that when they're writing (R, +, .) they mean that + and . represent some binary operations, and when they write (Z, +, .) they mean normal addition and multiplication ? (and in other cases I've to guess what they mean )

I still don't get it why they write (R, +, .) instead of (R, #, *). Writing # and * would vanish any doubt or confusion (at least mine). I'm having the same misunderstanding with more than one book.

Thank you very much for your help

 

[Mark44]

Like I already said in post #2, if the addition and multiplication operations are explicitly defined to be different, you can assume that they are the ordinary operations.