. . . . "Okruh endomorfism\u016F"@cs . . "Endomorphism ring"@en . . . . . . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u30A2\u30FC\u30D9\u30EB\u7FA4 X \u306E\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\uFF08\u82F1: endomorphism ring\uFF09End(X) \u306F\u3001X \u304B\u3089\u305D\u308C\u81EA\u8EAB\u3078\u306E\u6E96\u540C\u578B\u5199\u50CF\uFF08X \u4E0A\u306E\u81EA\u5DF1\u6E96\u540C\u578B\uFF09\u3059\u3079\u3066\u304B\u3089\u306A\u308B\u96C6\u5408\u3067\u3042\u308B\u3002\u52A0\u6CD5\u306F\uFF08\uFF09\u3067\u5B9A\u7FA9\u3055\u308C\u3001\u7A4D\u306F\u5199\u50CF\u306E\u5408\u6210\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u3002 \u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306E\u5143\u3068\u306A\u308B\u300C\u6E96\u540C\u578B\u300D\u304C\u4F55\u3092\u6307\u3059\u3082\u306E\u304B\u306F\u6587\u8108\u306B\u3088\u3063\u3066\u7570\u306A\u308A\u3001\u3053\u308C\u306F\u8003\u3048\u3066\u3044\u308B\u5BFE\u8C61\u306E\u570F\u306B\u4F9D\u5B58\u3059\u308B\u3002\u305D\u306E\u7D50\u679C\u3001\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306F\u5BFE\u8C61\u306E\u3044\u304F\u3064\u304B\u306E\u5185\u5728\u7684\u306A\u6027\u8CEA\u3092\u53D7\u3051\u7D99\u3044\u3067\u3044\u308B\u3002\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306F\u3057\u3070\u3057\u3070\u3042\u308B\u74B0\u4E0A\u306E\u591A\u5143\u74B0\uFF08\u4EE3\u6570\uFF09\u3067\u3042\u308A\u3001\u81EA\u5DF1\u6E96\u540C\u578B\u591A\u5143\u74B0\uFF08\u82F1: endomorphism algebra; \u81EA\u5DF1\u6E96\u540C\u578B\u4EE3\u6570\uFF09\u3068\u3082\u547C\u3070\u308C\u308B\u3002"@ja . . . . . . "Endomorphism ring"@en . . . "In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity. The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra. An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field F, its modules M are vector spaces V and their endomorphism rings are algebras over the field F."@en . "Endomorfiring"@sv . "Pier\u015Bcie\u0144 endomorfizm\u00F3w \u2013 pier\u015Bcie\u0144 skojarzony z pewnym rodzajem obiekt\u00F3w, kt\u00F3ry zawiera pewn\u0105 informacj\u0119 o jego w\u0142asno\u015Bciach wewn\u0119trznych."@pl . . . . . "In de algebra, een deelgebied van de wiskunde, bestaat de endomorfismenring van een abelse groep uit de endomorfismen van die groep. Deze endomorfismen vormen een ring, onder de elementsgewijze optelling en de functiecompositie als vermenigvuldiging."@nl . . . . . . "8828"^^ . . . . . "In de algebra, een deelgebied van de wiskunde, bestaat de endomorfismenring van een abelse groep uit de endomorfismen van die groep. Deze endomorfismen vormen een ring, onder de elementsgewijze optelling en de functiecompositie als vermenigvuldiging."@nl . . "Inom matematiken \u00E4r endomorfiringen av en abelsk grupp X, betecknad med End(X), m\u00E4ngden av alla homomorfier av X till sig sj\u00E4lv. Additionen definieras som punktvis addition av funktioner och multiplikationen definieras som funktionssammans\u00E4ttning."@sv . . . . . . . "Pier\u015Bcie\u0144 endomorfizm\u00F3w"@pl . . "p/e035610"@en . . . . . . "Pier\u015Bcie\u0144 endomorfizm\u00F3w \u2013 pier\u015Bcie\u0144 skojarzony z pewnym rodzajem obiekt\u00F3w, kt\u00F3ry zawiera pewn\u0105 informacj\u0119 o jego w\u0142asno\u015Bciach wewn\u0119trznych."@pl . . . . "Endomorfismenring"@nl . . . "Okruh endomorfism\u016F je matematick\u00E1 struktura z oboru abstraktn\u00ED algebry. Jej\u00EDmi prvky jsou endomorfismy n\u011Bjak\u00E9ho objektu (jin\u00E9 struktury) a dv\u011B operace \u2013 skl\u00E1d\u00E1n\u00ED endomorfism\u016F tohoto objektu, kter\u00E1 realizuje \u201En\u00E1soben\u00ED\u201C, a p\u016Fvodn\u00ED operace s\u010D\u00EDt\u00E1n\u00ED na objektu, p\u0159i\u010Dem\u017E v\u00FDsledn\u00E1 struktura spl\u0148uje axiomy okruhu. Nulov\u00FDm prvkem je endomorfismus zobrazuj\u00EDc\u00ED v\u0161e na nulov\u00FD prvek p\u016Fvodn\u00ED struktury a neutr\u00E1ln\u00EDm prvkem vzhledem k \u201En\u00E1soben\u00ED\u201C je identita. Okruh endomorfism\u016F b\u00FDv\u00E1 zna\u010Den End(X), kde X je nahrazeno ozna\u010Den\u00EDm p\u016Fvodn\u00ED struktury."@cs . . . "In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity."@en . . . . . . . . . . "\u81EA\u5DF1\u6E96\u540C\u578B\u74B0"@ja . "Okruh endomorfism\u016F je matematick\u00E1 struktura z oboru abstraktn\u00ED algebry. Jej\u00EDmi prvky jsou endomorfismy n\u011Bjak\u00E9ho objektu (jin\u00E9 struktury) a dv\u011B operace \u2013 skl\u00E1d\u00E1n\u00ED endomorfism\u016F tohoto objektu, kter\u00E1 realizuje \u201En\u00E1soben\u00ED\u201C, a p\u016Fvodn\u00ED operace s\u010D\u00EDt\u00E1n\u00ED na objektu, p\u0159i\u010Dem\u017E v\u00FDsledn\u00E1 struktura spl\u0148uje axiomy okruhu. Nulov\u00FDm prvkem je endomorfismus zobrazuj\u00EDc\u00ED v\u0161e na nulov\u00FD prvek p\u016Fvodn\u00ED struktury a neutr\u00E1ln\u00EDm prvkem vzhledem k \u201En\u00E1soben\u00ED\u201C je identita. Okruh endomorfism\u016F b\u00FDv\u00E1 zna\u010Den End(X), kde X je nahrazeno ozna\u010Den\u00EDm p\u016Fvodn\u00ED struktury."@cs . "1104382182"^^ . . "Inom matematiken \u00E4r endomorfiringen av en abelsk grupp X, betecknad med End(X), m\u00E4ngden av alla homomorfier av X till sig sj\u00E4lv. Additionen definieras som punktvis addition av funktioner och multiplikationen definieras som funktionssammans\u00E4ttning."@sv . . . . . . . . . . "59623"^^ . . . . . . . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u30A2\u30FC\u30D9\u30EB\u7FA4 X \u306E\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\uFF08\u82F1: endomorphism ring\uFF09End(X) \u306F\u3001X \u304B\u3089\u305D\u308C\u81EA\u8EAB\u3078\u306E\u6E96\u540C\u578B\u5199\u50CF\uFF08X \u4E0A\u306E\u81EA\u5DF1\u6E96\u540C\u578B\uFF09\u3059\u3079\u3066\u304B\u3089\u306A\u308B\u96C6\u5408\u3067\u3042\u308B\u3002\u52A0\u6CD5\u306F\uFF08\uFF09\u3067\u5B9A\u7FA9\u3055\u308C\u3001\u7A4D\u306F\u5199\u50CF\u306E\u5408\u6210\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u3002 \u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306E\u5143\u3068\u306A\u308B\u300C\u6E96\u540C\u578B\u300D\u304C\u4F55\u3092\u6307\u3059\u3082\u306E\u304B\u306F\u6587\u8108\u306B\u3088\u3063\u3066\u7570\u306A\u308A\u3001\u3053\u308C\u306F\u8003\u3048\u3066\u3044\u308B\u5BFE\u8C61\u306E\u570F\u306B\u4F9D\u5B58\u3059\u308B\u3002\u305D\u306E\u7D50\u679C\u3001\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306F\u5BFE\u8C61\u306E\u3044\u304F\u3064\u304B\u306E\u5185\u5728\u7684\u306A\u6027\u8CEA\u3092\u53D7\u3051\u7D99\u3044\u3067\u3044\u308B\u3002\u81EA\u5DF1\u6E96\u540C\u578B\u74B0\u306F\u3057\u3070\u3057\u3070\u3042\u308B\u74B0\u4E0A\u306E\u591A\u5143\u74B0\uFF08\u4EE3\u6570\uFF09\u3067\u3042\u308A\u3001\u81EA\u5DF1\u6E96\u540C\u578B\u591A\u5143\u74B0\uFF08\u82F1: endomorphism algebra; \u81EA\u5DF1\u6E96\u540C\u578B\u4EE3\u6570\uFF09\u3068\u3082\u547C\u3070\u308C\u308B\u3002"@ja . . . .