# rings & algebra

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Algebra. Algebra is the deep study of operations like multiplication, which turn two objects into a single object. This proof lies in the area of ring theory, which is concerned with the interaction of two such operations whose interaction is analogous to that of addition and multiplication. However, the “addition” and “multiplication” operations themselves are allowed to be rather exotic. For instance, this proof discusses a basic property of the “nilradical”. This is the set of nonzero elements, which have the counterintuitive property that, when multiplied by themself enough times, they give an answer of zero!

THIS IS BASICALLY THE EXACT OPPOSITE OF WHAT NORMALLY GETS ONE KICKED OUT OF A SCHOOL

Woman Mathematician part N

Amalie Emmy Noether – known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, Norbert Wiener and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras.

1 + 2 ≠ 3

• A plane is two-dimensional
• A line is one-dimensional
• A box is three-dimensional

But plane + line is not three-dimensional. 2 + 1 ≠ 3.

If one proves the equality of two numbers a and b by showing first that “a is less than or equal to b” and then “a is greater than or equal to b”, it is unfair, one should instead show that they are really equal by disclosing the inner ground for their equality.
—

Emmy Noether

“Beneath the effort directed toward the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.
In the judgment of the most competent living mathematicians,
Fraeulein Noether was the most significant creative mathematical genius thus far produced since higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of
enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the
largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulae are discovered necessary for the deeper penetration into the laws of nature. “Albert Einstein - Here is a story of what Albert Einstein said of Emmy Noether .

Emmy Noether was born on March 23, 1882 and died on April 14, 1935. was a mathematician known for her landmark contributions to abstract algebra and theoretical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, Noether’s theorem explains the connection between symmetry and conservation laws.

If you understand this, you’re a genius because I don’t understand it and I wrote it.
—  Ring theory professor

## Emmy Noether and Noether’s First Theorem

Today is the birthday of German Mathematician and physicist Emmy Noether, born March 23, 1882 in Erlangen, died at 53 in Bryn Mawr Pennsylvania on April 14, 1935.  Among her many discoveries, Noether’s First Theoem turns 100 years old this year.  Noether’s First theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.  The tumbling asteroid Toutatis is a perfect example of this.  Toutatis is considered a very odd asteroid-not only is its shape complex, but it tumbles through space, unlike most asteroids that spiral like a well tossed football.  As an explanation of Noether’s First Theorem, Wikipedia offers:

As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric: from this symmetry, Noether’s theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry — it is the laws of its motion that are symmetric.

Emmy Noether worked on many complex problems in her short life and contributed mightily to physics, algebra, ring theory.  She fled Nazi Germany in 1933 but died of ovarian cancer shortly after arriving in the United States.

GIF of Toutatis courtesy Kasia Cieplak-Mayr von Baldegg via The Atlantic Magazine.  Quotation courtesy Wikipedia used with permission under a Creative Commons 3.0 license.

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my illustration of the first isomorphism theorem, which says you can replace an arrow `ƒ:X→Y` by a sequence of arrows `surjection ∘ bijection ∘ injection`.

The Lord of the Rings :D

Diagram of ring types/properties

Nifty!

I didn’t really find out why (-1)(-1)=1 until I started studying ring theory.
—  Abstract Algebra Lecturer

Emmy Noether (official name Amalie Emmy Noether; 23 March 1882 – 14 April 1935), was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, Norbert Wiener and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether’s theorem explains the fundamental connection between symmetry and conservation laws.

Most of the time, 0 is not equal to 1. That is not the case here.
—  Abstract algebra professor discussing the 0 ring