- #1
LikeMath
- 62
- 0
Hey!
Is there an infinite ring with exactly two maximal ideals.
Thanks in advance
LiKeMath
Is there an infinite ring with exactly two maximal ideals.
Thanks in advance
LiKeMath
How is multiplication defined here?micromass said:What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?
Erland said:How is multiplication defined here?
An infinite ring with exactly two non trivial maximal ideals is a ring that contains an infinite number of elements and has exactly two non-trivial maximal ideals. A maximal ideal is a proper subset of the ring that cannot be properly contained in any other ideal. Non-trivial maximal ideals have elements other than 0 and the entire ring.
One example is the ring of Gaussian integers, which contains all complex numbers of the form a + bi, where a and b are integers and i is the imaginary unit. Another example is the ring of polynomials over a field, such as the ring of all polynomials with real coefficients.
Infinite rings with exactly two non trivial maximal ideals are different from other rings because they have a unique structure and properties that are not shared by other rings. They also have a unique number of maximal ideals, which affects the behavior of elements in the ring.
No, an infinite ring with exactly two non trivial maximal ideals can only have two maximal ideals. This is because the definition of a maximal ideal states that it cannot be properly contained in any other ideal, and an infinite ring with exactly two non trivial maximal ideals by definition only has two maximal ideals.
The study of infinite rings with exactly two non trivial maximal ideals has applications in abstract algebra, which has applications in many areas of mathematics and science. It also has applications in number theory, cryptography, and coding theory. Understanding the structure and properties of these rings can also lead to further developments in these areas.