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Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the ...
Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. [12] Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
The ideal A is a prime ideal in R if all but one of the A i are equal to R i and the remaining A i is a prime ideal in R i. However, the converse is not true when I is infinite. For example, the direct sum of the R i form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a ...
In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal = is especially important, for example the distinguished maximal ideal of a valuation ring.
Over the last 25 years, (ED) medications such as Viagra and others have become common and normal pieces of bedroom tool kit. These little pills have helped hundreds of millions of men all over the ...
For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the ...
Any maximal ideal in a commutative ring is a quotient von Neumann regular ideal since R/M is a field. This is not true in general because for noncommutative rings R/M may only be a simple ring, and may not be von Neumann regular. Let R be a local ring which is not a division ring, and with maximal right ideal M.
Myofascial release (MFR, self-myofascial release) is an alternative medicine therapy claimed to be useful for treating skeletal muscle immobility and pain by relaxing contracted muscles, improving blood and lymphatic circulation and stimulating the stretch reflex in muscles.