nil ideal造句

• In a right artinian ring, any nil ideal is nilpotent.
• Nil ideals are still associated with interesting open questions, especially the unsolved K鰐he conjecture.
• In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings.
• In the case of commutative rings, there is always a maximal nil ideal : the nilradical of the ring.
• The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings.
• The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil.
• The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil.
• He began working in ring theory and in 1930 published the K鰐he conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal.
• He began working in ring theory and in 1930 published the K鰐he conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal.
• The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil.
• However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the K鰐he conjecture.
• However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the K鰐he conjecture.
• This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal ( due to the artinian hypothesis ), the result follows.
• For associative rings, the definition of Zorn ring can be restated as follows : the Jacobson radical J ( " R " ) is a nil ideal and every right ideal of " R " which is not contained in J ( " R " ) contains a nonzero idempotent.