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A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring: For any two ideals A and B of R, AB = {0} implies A = {0} or B = {0}.
Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t.
In a commutative ring R with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal (0) is prime, then the ring R is an integral domain. If q is any non-zero element of R and the ideal (q 2) is prime, then it contains q and then q is invertible.)
This extends the definition for commutative rings. 4. prime ring : A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every simple ring and every domain is a prime ring. primitive 1.
In fact, every non-zero ideal of the ring is generated by its smallest positive element, as a consequence of Euclidean division, so is a principal ideal domain. [ 9 ] The set of all polynomials with real coefficients that are divisible by the polynomial x 2 + 1 {\displaystyle x^{2}+1} is an ideal in the ring of all real-coefficient polynomials ...
The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The ...
The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no nonzero zero divisors. A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.
An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b. With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers .