Thank you for understanding your material so well that you are able to explain it in a way that helps others also understand. You are a true example of a teacher.
Many ring theorists today include the existence of a multiplicative identity as part of the definition of a ring. There has been a perspective shift from more abstract branches of mathematics, in particular universal algebra and category theory, which show us that having a multiplicative identity should be seen as the standard, and not having a multiplicative identity should be seen as a sort of "something is missing" case. Rings without identity can be referred to as "rngs".
Hi One of my friends pointed out this interesting thing. Commutativity of the Addition in a ring comes as a consequence (of Distributivity) rather than an axiom in the definition of a ring. This can be shown by considering (1 + 1) * (x + y) in two different ways. (1 + 1) * (x + y) = (1 + 1) * x + (1 + 1) * y ...by Left Distributivity = (1 * x + 1 * x) + (1 * y + 1 * y) ...by Right Distributivity = (x + x) + (y + y) ...as 1 is Multiplicative Identity = x + (x + y) + y ...by Associativity of Addition (1 + 1) * (x + y) = 1 * (x + y) + 1 * (x + y) ...by Right Distributivity = (x + y) + (x + y) ...as 1 is Multiplicative Identity = x + (y + x) + y ...by Associativity of Addition This gives x + (x + y) + y = x + (y + x) + y That is x + y = y + x The takeaway is that we can ask for a smaller number of axioms to hold in order to get the same structure of a ring. Food for thought !
You are absolutely correct. The same is true for the definition of a vector space and the definition of a module. Some texts do _not_ require a 1 element to be in a ring, however. If the 1 element is not required, then you need to include the commutativity of addition as an axiom. Requiring 1 to be in a ring is a relatively newer thing, which explains why it remains as an axiom today.
Just a small note: I don't think a ring strictly needs to have a multiplicative identity. In my textbook, they call a ring that has a multiplicative identity a "ring with unity".
This is a possible definition. However, most ring theorists today believe that rings "should" have a multiplicative identity. As such, it is commonplace now to require the existence of a multiplicative identity in the definition of a ring. Rings without identity are then often referred to as "rngs" (losing the i corresponding to losing the identity), since they are missing something that they "should" have.
yes, take for instance the integers with usual addition and multiplication. Most of the elements won't have a multiplicative inverse in the integers, but it's still a ring
@@stinkenderig Thanks, this example clarified it for me. So a ring is basically a structure in which you can add, subtract and multiply, but you can't divide yet? (because there are additive inverses, but not multiplicative inverses)
(A,*) has a structure which is called a "monoid". It's actually a decently important algebraic structure in higher mathematics. A monoid is, essentially, a group without the inverse requirement. The binary operation is associative, closed, and has an identity element, but need not have inverses. And that's what's going on with (A,*). :)
Thank you for understanding your material so well that you are able to explain it in a way that helps others also understand. You are a true example of a teacher.
how do we pay you for this
The presence of a multiplicative identity is NOT a requirement for a general ring!
Many ring theorists today include the existence of a multiplicative identity as part of the definition of a ring. There has been a perspective shift from more abstract branches of mathematics, in particular universal algebra and category theory, which show us that having a multiplicative identity should be seen as the standard, and not having a multiplicative identity should be seen as a sort of "something is missing" case.
Rings without identity can be referred to as "rngs".
Hi
One of my friends pointed out this interesting thing.
Commutativity of the Addition in a ring comes as a consequence (of Distributivity) rather than an axiom in the definition of a ring.
This can be shown by considering (1 + 1) * (x + y) in two different ways.
(1 + 1) * (x + y)
= (1 + 1) * x + (1 + 1) * y ...by Left Distributivity
= (1 * x + 1 * x) + (1 * y + 1 * y) ...by Right Distributivity
= (x + x) + (y + y) ...as 1 is Multiplicative Identity
= x + (x + y) + y ...by Associativity of Addition
(1 + 1) * (x + y)
= 1 * (x + y) + 1 * (x + y) ...by Right Distributivity
= (x + y) + (x + y) ...as 1 is Multiplicative Identity
= x + (y + x) + y ...by Associativity of Addition
This gives x + (x + y) + y = x + (y + x) + y
That is x + y = y + x
The takeaway is that we can ask for a smaller number of axioms to hold in order to get the same structure of a ring.
Food for thought !
You are absolutely correct. The same is true for the definition of a vector space and the definition of a module.
Some texts do _not_ require a 1 element to be in a ring, however. If the 1 element is not required, then you need to include the commutativity of addition as an axiom. Requiring 1 to be in a ring is a relatively newer thing, which explains why it remains as an axiom today.
Just a small note: I don't think a ring strictly needs to have a multiplicative identity. In my textbook, they call a ring that has a multiplicative identity a "ring with unity".
This is a possible definition. However, most ring theorists today believe that rings "should" have a multiplicative identity. As such, it is commonplace now to require the existence of a multiplicative identity in the definition of a ring.
Rings without identity are then often referred to as "rngs" (losing the i corresponding to losing the identity), since they are missing something that they "should" have.
@@MuffinsAPlenty Makes sense. Thanks
A ring with an identity crisis :)
So where's the inverses axiom for multiplication? Since it's missing, does it mean that multiplication in a ring doesn't form a group? :q
yes, take for instance the integers with usual addition and multiplication. Most of the elements won't have a multiplicative inverse in the integers, but it's still a ring
@@stinkenderig Thanks, this example clarified it for me.
So a ring is basically a structure in which you can add, subtract and multiply, but you can't divide yet? (because there are additive inverses, but not multiplicative inverses)
what about Division ring and zero divisor of a ring
(A,+) its an Abelian Group, (A,*) its only associative
(A,*) has a structure which is called a "monoid". It's actually a decently important algebraic structure in higher mathematics. A monoid is, essentially, a group without the inverse requirement. The binary operation is associative, closed, and has an identity element, but need not have inverses. And that's what's going on with (A,*). :)