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Under a proposed order, which must be approved by a federal court before it 
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also will be required to implement a privacy and security program with 
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security controls, such as multi-factor authentication for both employee 
and customer accounts.

The Commission voted 3-0 to authorize the staff to file the complaint and 
stipulated final order. The FTC filed the complaint and final order in the 
U.S. District Court for the District of the District of Columbia.

The Federal Trade Commission works to promote competition and protect and 
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In mathematics, *rings* are algebraic structures that generalize fields: 
multiplication need not be commutative and multiplicative inverses need not 
exist. Informally, a *ring* is a set equipped with two binary operations 
satisfying properties analogous to those of addition and multiplication of 
integers. Ring elements may be numbers such as integers or complex numbers, 
but they may also be non-numerical objects such as polynomials, square 
matrices, functions, and power series.

Formally, a *ring* is a set endowed with two binary operations called 
*addition* and *multiplication* such that the ring is an abelian group with 
respect to the addition operator, and the multiplication operator is 
associative, is distributive over the addition operation, and has a 
multiplicative identity element. (Some authors define rings without 
requiring a multiplicative identity and instead call the structure defined 
above a *ring with identity*. See * Variations on the definition*.)

Whether a ring is commutative has profound implications on its behavior. 
Commutative algebra, the theory of commutative rings, is a major branch of 
ring theory. Its development has been greatly influenced by problems and 
ideas of algebraic number theory and algebraic geometry. The simplest 
commutative rings are those that admit division by non-zero elements; such 
rings are called fields.

The conceptualization of rings spanned the 1870s to the 1920s, with key 
contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first 
formalized as a generalization of Dedekind domains that occur in number 
theory, and of polynomial rings and rings of invariants that occur in 
algebraic geometry and invariant theory. They later proved useful in other 
branches of mathematics such as geometry and analysis.

Although ring addition is commutative, ring multiplication is not required 
to be commutative: ab need not necessarily equal *ba*. Rings that also 
satisfy commutativity for multiplication (such as the ring of integers) are 
called *commutative rings*. Books on commutative algebra or algebraic 
geometry often adopt the convention that *ring* means *commutative ring*, 
to simplify terminology.

The additive group of a ring is the underlying set equipped with only the 
operation of addition. Although the definition requires that the additive 
group be abelian, this can be inferred from the other ring axioms.[4] The 
proof makes use of the "1", and does not work in a rng. (For a rng, 
omitting the axiom of commutativity of addition leaves it inferable from 
the remaining rng assumptions only for elements that are products: *ab* + 
*cd* = *cd* + *ab*.)

The study of rings originated from the theory of polynomial rings and the 
theory of algebraic integers.[11] In 1871, Richard Dedekind defined the 
concept of the ring of integers of a number field.[12] In this context, he 
introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal 
number) and "module" and studied their properties. Dedekind did not use the 
term "ring" and did not define the concept of a ring in a general setting.

The first axiomatic definition of a ring was given by Adolf Fraenkel in 
1915,[15][16] but his axioms were stricter than those in the modern 
definition. For instance, he required every non-zero-divisor to have a 
multiplicative inverse.[17] In 1921, Emmy Noether gave a modern axiomatic 
definition of commutative rings (with and without 1) and developed the 
foundations of commutative ring theory in her paper *Idealtheorie in 
Ringbereichen*.[18]

Most or all books on algebra[20][21] up to around 1960 followed Noether's 
convention of not requiring a 1 for a "ring". Starting in the 1960s, it 
became increasingly common to see books including the existence of 1 in the 
definition of "ring", especially in advanced books by notable authors such 
as Artin,[22] Bourbaki,[23] Eisenbud,[24] and Lang.[3] There are also books 
published as late as 2022 that use the term without the requirement for a 
1.[25][26][27][28] Likewise, the Encyclopedia of Mathematics does not 
require unit elements in rings.[29] In a research article, the authors 
often specify which definition of ring they use in the beginning of that 
article.

Gardner and Wiegandt assert that, when dealing with several objects in the 
category of rings (as opposed to working with a fixed ring), if one 
requires all rings to have a 1, then some consequences include the lack of 
existence of infinite direct sums of rings, and that proper direct summands 
of rings are not subrings. They conclude that "in many, maybe most, 
branches of ring theory the requirement of the existence of a unity element 
is not sensible, and therefore unacceptable."[30] Poonen makes the 
counterargument that the natural notion for rings would be the direct 
product rather than the direct sum. However, his main argument is that 
rings without a multiplicative identity are not totally associative, in the 
sense that they do not contain the product of any finite sequence of ring 
elements, including the empty sequence.[c][31]

A nilpotent element is an element a such that *an* = 0 for some *n* > 0. 
One example of a nilpotent element is a nilpotent matrix. A nilpotent 
element in a nonzero ring is necessarily a zero divisor.

An intersection of subrings is a subring. Given a subset E of R, the 
smallest subring of R containing E is the intersection of all subrings of R 
containing E, and it is called *the subring generated by E*.

If x is in R, then *Rx* and *xR* are left ideals and right ideals, 
respectively; they are called the principal left ideals and right ideals 
generated by x. The principal ideal *RxR* is written as (*x*). For example, 
the set of all positive and negative multiples of 2 along with 0 form an 
ideal of the integers, and this ideal is generated by the integer 2. In 
fact, every ideal of the ring of integers is principal.
4a15465005

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