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# Topic: Boolean ring

###### In the News (Thu 23 May 13)

 NationMaster - Encyclopedia: Boolean ring A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal. Boolean rings are equivalent to Boolean algebras (or Boolean lattices). www.nationmaster.com /encyclopedia/Boolean-ring   (523 words)

 PlanetMath: Boolean ring As mentioned above, every Boolean algebra can be considered as a Boolean ring. forms a Boolean ring, with intersection as multiplication and symmetric difference as addition. This is version 20 of Boolean ring, born on 2002-02-24, modified 2006-08-03. www.planetmath.org /encyclopedia/BooleanRing.html   (128 words)

 Boolean algebra - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-10-15) The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a for all a in A; rings with this property are called Boolean rings. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. encyclopedia.worldsearch.com /boolean_algebra.htm   (1565 words)

 Ring (mathematics) - Wikipedia, the free encyclopedia In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. A ring (in the categorical sense) is commutative iff it is equal to its opposite ring. The split-complex plane D is a ring useful in modern physics and is a subring of the tessarines. en.wikipedia.org /wiki/Ring_(mathematics)   (1102 words)

 Boolean ring - Wikipedia, the free encyclopedia A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal. en.wikipedia.org /wiki/Boolean_ring   (507 words)

 Learn more about Boolean algebra in the online encyclopedia.   (Site not responding. Last check: 2007-10-15) The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm). Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, *) by defining a + b = (a ∧ ¬b) ∨ (b ∧ ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a * b = a ∧ b. An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I. www.onlineencyclopedia.org /b/bo/boolean_algebra.html   (1669 words)

 Rings The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. www.risberg.ws /Hypertextbooks/Mathematics/Algebra/rings.htm   (890 words)

 [No title] A Boolean ring is defined as a ring with a multiplicative identity where x2 = x for all x in R. This reveals two important facts. In fact, all prime ideals p in a Boolean ring are maximal ideals, meaning that the ideal is proper and it is not contained in any other proper ideal. Since R/p is an integral domain and a Boolean ring, it is isomorphic to the field F2, and therefore p is maximal. www.facstaff.bucknell.edu /pbrooksb/320/lianne.doc   (392 words)

 boolean ring   (Site not responding. Last check: 2007-10-15) In mathematics, a Boolean ring R is a ring for which x is a Boolean ring: consider for instance the polynomial ring F Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and at the same time a Boolean ring, so it must be isomorphic to the field F www.yourencyclopedia.net /Boolean_ring   (465 words)

 Boolean Ring Thus boolean logic involves variables that are true or false, and boolean circuitry, which drives your computer and mine, is based on solid state switches that are either on or off. A boolean ring that is finite, or a finite dimensional Z This notation is concise, and it reflects the bitwise operations that are typical of a boolean lattice. www.mathreference.com /ring-jr,boolring.html   (1228 words)

 Math 441 Hw 19 Give an example of a ring R and elements a,b, and c in R such that A Boolean ring is a ring with the property that Show that a Boolean ring is a commutative ring with www.andrews.edu /~ohy/math441/Hw19.htm   (46 words)

 The Mathematics of Boolean Algebra Boolean algebra is the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation. The study of Boolean algebras has several aspects: structure theory, model theory of Boolean algebras, decidability and undecidability questions for the class of Boolean algebras, and the indicated applications. Much of the deeper theory of Boolean algebras, telling about their structure and classification, can be formulated in terms of certain functions defined for all Boolean algebras, with infinite cardinals as values. plato.stanford.edu /entries/boolalg-math   (2064 words)

 Boolean logic Article, Booleanlogic Information   (Site not responding. Last check: 2007-10-15) The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involvingseveral variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (whichcan always be checked by a trivial brute force algorithm). The zero element of thisring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Booleanalgebra. This ring has the property that a * a = a for all a in A; rings with thisproperty are called Boolean rings. www.anoca.org /algebra/element/boolean_logic.html   (1453 words)

 [No title]   (Site not responding. Last check: 2007-10-15) Rings are things like the integers, whose addition, subtraction and multiplication behave in pretty much the usual way. As I understand it, you're observing that Boolean algebras are defined differently from rings, and you're asking about a connection between them. (This is called a Boolean ring.) Now define meet (intersection) and join (union) operations by: x meet y = xy, x join y = xy + x + y. www.math.niu.edu /~rusin/known-math/99/boolean_ring   (394 words)

 ring   (Site not responding. Last check: 2007-10-15) Lord of the Rings: The Fellowship of the... In ancient times the Rings of Power were crafted by the Elven-smiths, and Sauron, the Dark Lord, forged the One Ring, filling it with his own power so... Lord of the Rings: Fellowship of the Rin... www.byglrb.com /jewelry/ring&start=90   (1416 words)

 No Title   (Site not responding. Last check: 2007-10-15) Let A be a commutative ring and I an ideal distinct from A. Deduce that in a Boolean ring every prime ideal is maximal. Let A be a principal ring and S a multiplicative set. www.math.gatech.edu /~saugata/teaching/fall00/assignment5/assignment5.html   (204 words)

 [No title] In mathematics, a Boolean ring R is a Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an en-cyclopedia.com /wiki/Boolean_ring   (386 words)

 n_boolean A Boolean ring is a ring in which every element is idempotent. Any Boolean ring with 1 can be made into a Boolean algebra by defining , a subset is a lattice ideal if and only if it is a ring ideal with respect to the resulting ring structure. www.math.ucla.edu /~baker/222a/handouts/n_boolean/node4.html   (75 words)

 [No title] More generally, an atom is minimal, non-zero element in a Boolean algebra. Boolean algebra +------------------------------------------------------------ A Boolean algebra is a set S with two binary operations + and * which are commutative monoids (S,+,0), (S,*,1) and satisfy the two distributive laws (x*(y+z)=x*y + y*z, x+(y*z) =(x+y)*(x+z) as well as the complementary laws x*x=1, y+y=0. Boolean ring +------------------------------------------------------------ A Boolean ring is a ring in which every member is idempotent. www.math.harvard.edu /~knill/sofia/data/measuretheory.txt   (702 words)

 M567: Boolean Algebra of ideals of the Boolean algebra B is not a Boolean algebra. Show that a subset of a Boolean algebra is an ideal of the Boolean algebra if and only if it is an ideal of the corresponding Boolean ring. Show that a subset of a Boolean algebra is a prime ideal of the Boolean algebra if and only if it is a prime ideal of the corresponding Boolean ring. orion.math.iastate.edu /jdhsmith/class/M567S05.htm   (880 words)

 ► » Re: Embedding Boolean Algebras   (Site not responding. Last check: 2007-10-15) Boolean rings (i.e., commutative rings with identity in which Boolean ring A^ of a Boolean algebra A to have the same I is maximal ideal of Boolean ring R^ Please answer this yes-or-no question: www.science-chat.org /Re-Embedding-Boolean-Algebras-6936459.html   (965 words)

 The Spectrum of a Boolean Ring   (Site not responding. Last check: 2007-10-15) Spec R, The Spectrum of a Boolean Ring Stone's theorem states that any boolean lattice is the collection of clopen sets in some compact hausdorff topological space, with meet being intersection and join being union. Convert the boolean lattice to a ring, and then derive spec R. The clopen sets in spec R correspond to the elements of R, which are the elements of the lattice. www.mathreference.com /ring-zar,bool.html   (340 words)

 [No title]   (Site not responding. Last check: 2007-10-15) However this is not the case in the infinite setting (i.e., there are Boolean Algebras which are not powerset algebras). M.H. Stone though gave a classification of (infinite) Boolean Algebras when he in 1936 proved that every Boolean Algebra is the algebra of clopen subsets of a compact hausdorff space. This talk will attempt to present this theorem by associating to a boolean algebra a commutative ring called a boolean ring and looking at the Zariski topology over its prime spectrum. www.math.uconn.edu /sigma/S05_3.htm   (112 words)

 Foster   (Site not responding. Last check: 2007-10-15) His work on generalising Boolean rings culminated in his classic paper The theory of Boolean-like rings which he published in the Transactions of the American Mathematical Society in 1946. A "Boolean-like" ring is a commutative ring H with unit element such that a + a = 0 for all a in H and ab(a + b + ab) = ab for any two elements a, b of H. The main properties of such rings are Foster went on to define the concept of a primal algebra generalising a Boolean algebra within the theory of varieties of universal algebras. www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Foster.html   (1238 words)

 : Class Ring Ring represents a circular 1 dimensional space bound by 0 to 2^dimension. Nodes and Keys are added to the ring using a simple hash function that needs to be implemented using the consistent hashing algorithm of D.Karger. Adds a node to the ring by assigning an identifier to the node (using a hash of its IP) as well as filling this "ring spot" with this node. www.caip.rutgers.edu /~vincentm/CHORD/DOCS/Ring.html   (426 words)

 Lloyd's Lounge: Wednesday, March 24th, 12:46 PM A Boolean Ring is a ring in which a^2 = a, so named in Honor of the English mathematician George Boole (1815-1864). Boolean Algebra, as defined by De Morgan's theorems, is consistently capitalized, again, in the possesive case. These Boolean operations are the heart of many CAD/CAM software systems, which, for example, construct complex parts for numerically controlled machining by subtracting one shape from another, joining shapes, slicing away part of a shape, ect., all of which are Boolean operations." (Computational Geometry in C, O'Rourke, 2nd ed, 2000, pp 268)... www.lloydslounge.org /entry.php?id=1080161177   (826 words)

 Exercises 3 Let R be the Boolean ring of Exercises 2 Question 1. If R is a commutative ring with identity whose only ideals are {0} and R, prove that R is a field. Prove that R is a subring of the ring of all real matrices. www-groups.dcs.st-and.ac.uk /~john/MT4517/Tutorials/T3.html   (375 words)

 Exercises 2 Show that the set of all subsets of a set S forms a ring under the operations A + B = A Let R be the ring of elements of the form {a + bx Prove that R is a ring with p www-groups.dcs.st-and.ac.uk /~john/MT4517/Tutorials/T2.html   (296 words)

 SET THEORY, QUANTUM SET THEORY & CLIFFORD ALGEBRAS Closure on bivectors with vectors on ISO(n) ring A Boolean algebra is not at all what one would call an "algebra" in the modern sense of a vector space over a field with a multiplication defined as a binary operation on the vectors. A ring is a collection R of {A, B,...} (not necessarily implying even countability) on which two operations, x and + are defined so that: 1. graham.main.nc.us /~bhammel/QSET/qset1.html   (10449 words)

 [No title]   (Site not responding. Last check: 2007-10-15) % This input file uses Boolean ring rules to rewrite % Boolean expressions into canonical form in terms of % {and,xor,0,1}. All constants in the expressions to be rewritten % must occur in the lex command. % The rest of the rules rewrite Boolean ring terms into canonical form. www-unix.mcs.anl.gov /AR/otter/examples/fringe/bring.in   (371 words)

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