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Elementary mathematics, also known as primary or secondary school mathematics, is the study of mathematics topics that are commonly taught at the primary or secondary school levels around the world. It includes a wide range of mathematical concepts and skills, including number sense , algebra , geometry , measurement , and data analysis .
Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry , a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X : heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theory also studies the natural, or whole, numbers.
[1] Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation.
[7] [8] Some districts in other states, including California, have either switched or are considering switching to an integrated curriculum. [9] Under the Common Core Standards adopted by most states in 2012, high school mathematics may be taught using either a traditional American approach or an integrated curriculum. The only difference would ...
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N 1 and N 2 are submodules of M such that + =, then either N 1 = M or N 2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule .
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set .