Podcast for Ring Theory History here!
Podcast Script Below!
In this video, we will be taking a closer inspection of the history of Ring Theory and who invented the axioms we know today.
The history of Ring Theory goes back to 1914, when German mathematician Abraham Fraenkel provided the first axiomatic definition of a ring in his book, “Uber die Teiler der Null und die Zerlegung von Ringen.
”. However, despite being the first to derive axiomatic definitions for rings, Fraenkel’s axioms are not the modern axioms we known today. In particular, Fraenkel’s were much stricter, with one of the notable
differences being that he required non-zero divisors to have a multiplicative inverse.
So, who can we attribute the modern axioms of Ring Theory to? Well, the modern axiomatic definition of a ring comes from Amalie Emmy Noether (A-ma-li-eh emmy new-ta), another German mathematician who, in 1921,
wrote a book called “Idealtheorie in Ringbereichen (I-de-al-tear-ee in Ring-ber-i-hen). Noether gave the modern definition of a commutative ring. A commutative ring is defined as a ring that satisfies
commutative multiplication. In modern times and thus what you will see on the rest of this webpage, a ring has 8 axioms, with a commutative ring having 9. These axioms, in very simple terms, are the following:
1. Closure under addition
2. Associative addition
3. Commutative addition
4. An Additive identity
5. An Additive Inverse
6. Closure under multiplication
7. Associative multiplication
And for the 8th one:
8. The various Distributive Laws
These are the basic Ring Axioms. To make a commutative ring though, we just have to add one more to the basic 8
9. Commutative Multiplication
The big split between Fraenkel and Noether was whether to include a multiplicative identity as part of the definition of a ring. Fraenkel included it, but Noether did not. Today, there are books that adopt Fraenkel’s
belief and some which adopt Noether’s. On this webpage, we will be using Noether’s axioms as the basis. In our list of axioms, you will find that a “ring with identity” is a special kind of ring which includes this
multiplicative identity as an axiom, however, it isn’t part of our base 8.
And this is a brief history of how the Ring Theory axioms came to be. Thanks for watching, and please check out rest of the site for more info on Ring Theory and Fields!
Podcast Script Below!
In this video, we will be taking a closer inspection of the history of Ring Theory and who invented the axioms we know today.
The history of Ring Theory goes back to 1914, when German mathematician Abraham Fraenkel provided the first axiomatic definition of a ring in his book, “Uber die Teiler der Null und die Zerlegung von Ringen.
”. However, despite being the first to derive axiomatic definitions for rings, Fraenkel’s axioms are not the modern axioms we known today. In particular, Fraenkel’s were much stricter, with one of the notable
differences being that he required non-zero divisors to have a multiplicative inverse.
So, who can we attribute the modern axioms of Ring Theory to? Well, the modern axiomatic definition of a ring comes from Amalie Emmy Noether (A-ma-li-eh emmy new-ta), another German mathematician who, in 1921,
wrote a book called “Idealtheorie in Ringbereichen (I-de-al-tear-ee in Ring-ber-i-hen). Noether gave the modern definition of a commutative ring. A commutative ring is defined as a ring that satisfies
commutative multiplication. In modern times and thus what you will see on the rest of this webpage, a ring has 8 axioms, with a commutative ring having 9. These axioms, in very simple terms, are the following:
1. Closure under addition
2. Associative addition
3. Commutative addition
4. An Additive identity
5. An Additive Inverse
6. Closure under multiplication
7. Associative multiplication
And for the 8th one:
8. The various Distributive Laws
These are the basic Ring Axioms. To make a commutative ring though, we just have to add one more to the basic 8
9. Commutative Multiplication
The big split between Fraenkel and Noether was whether to include a multiplicative identity as part of the definition of a ring. Fraenkel included it, but Noether did not. Today, there are books that adopt Fraenkel’s
belief and some which adopt Noether’s. On this webpage, we will be using Noether’s axioms as the basis. In our list of axioms, you will find that a “ring with identity” is a special kind of ring which includes this
multiplicative identity as an axiom, however, it isn’t part of our base 8.
And this is a brief history of how the Ring Theory axioms came to be. Thanks for watching, and please check out rest of the site for more info on Ring Theory and Fields!