Basic Properties of ring I have just finished reading groups and move to rings . I have been thinking rings like a group with some extra features .Like for addition ring is a good abelian group and for multiplication it has some more or less properties . But I have some doubts ( Pardon me if these questions seems silly )

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*Can addition and multipication be any binary operation in ring ? Are '+ 'and '• 'are just symbols replacing Asterisk for any binary operation in  groups .
Can addition be multipication and multipication be addition .

Is $a^2=a.a$ or $a+a$ ?


*How can we prove that ring is closed for multiplication process ( Closure property ) . Because in definition it only follows associativity and distribution ?

 A: When you give the definition of group you don't "prove" it is closed under multiplication, that's what happens if you have to show a certain set with an operation is indeed a group. Same thing for rings, there's no proof needed. You should observe that a binary operation on a set $A$ is a function $f: A\times A\longrightarrow A$ with some extra properties, thus closure is already assured by definition.
Generally in group theory we use the term "operation" and often denote the operation as "$\star$". But now in ring theory as you have $2$ operations, you need to give them a more proper name, hence $+$ and $\cdot$ come into play. Please notice "$-$" is not a binary operation, but rather a unary operation associating to an element its inverse. Writing $a-b$ is just a short form for $a+(-b)$.
The notation $a^{2}$ can be tricky sometimes. In a group this would depend on the operation you have, in a ring it always means $a\cdot a$ and not $a+a$.
A: *

*They have one main constraint, they have to be distributive.
for every three elements in the ring, $a,b,c$, $(a+b)*c=ac+bc$.
one of the main reasons multiplication can't be addition is because 0, it is rather easy to prove that multiplication by 0 is absorptive($0*a = (1-1)*a = a-a=0$).
but this conflicts with it being a group, as it has to have an identity.

