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Let R be a non-commutative ring . Suppose that the number of non-units of R is finite . Can we say that R is a finite ring?
zhentil said:If R and F are rings, then RxF is a ring. In particular, if R is a finite non-commutative ring and F is a field...
Landau said:then RxF is an infinite non-commutative ring...
so...
The finiteness of a non-commutative ring refers to whether or not the ring has a finite number of elements. A non-commutative ring is considered finite if it has a finite number of elements, and infinite if it has an infinite number of elements.
Studying the finiteness of non-commutative rings is important for understanding and classifying these types of rings. It also has practical applications in areas such as algebraic geometry and coding theory.
The finiteness of a non-commutative ring can be determined by examining its structure and properties. If the ring has a finite number of elements, it can be proven using mathematical techniques such as induction or contradiction.
Some examples of finite non-commutative rings include the ring of 2x2 matrices over a finite field, the ring of quaternions, and the ring of square matrices over a finite field.
No, a non-commutative ring cannot be both finite and infinite. It must fall into one of these two categories based on the number of elements it contains.