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For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
A ring with property r is called an r-ring; an ideal of some ring with property r is called an r-ideal. In particular, the r-ideals are a subset of the r-rings. A ring is said to be a r-semi-simple ring if it has no non-zero r-ideals. r is said to be a radical property if: the class of r-rings is closed under homomorphic images
The Ring of Fire (also known as the Pacific Ring of Fire, the Rim of Fire, the Girdle of Fire or the Circum-Pacific belt) [note 1] is a tectonic belt of volcanoes and earthquakes. It is about 40,000 km (25,000 mi) long [ 1 ] and up to about 500 km (310 mi) wide, [ 2 ] and surrounds most of the Pacific Ocean .
A total solar eclipse is far different from a partial eclipse or a ring of fire event, as the moon completely covers the sun, casting a shadow that plunges a swath of the Earth into darkness for ...
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Consider the ring of integers.. The radical of the ideal of integer multiples of is (the evens).; The radical of is .; The radical of is .; In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer).
A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
The "ring of fire" eclipse is pictured in Puerto San Julian, Santa Cruz province, Argentina on October 2, 2024. Where was the solar eclipse visible? Southern parts of Argentina and Chile sew the ...