DEFINITION. Let G be a group and K a commutative ring
On modules over group rings of soluble groups with commutative ring of scalars
... 9 G includes a finitely generated subgroup S such that G/S is a Pr¨ ufer p-group for some prime ...Then G = M × K, where M is a Pr¨ ufer p-group for some prime p, K is a ...
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(G) of the cat- egory of G-equivariant C-vector bundles on G (for the action of G on itself by conjugation) is endowed with a structure of (commutative) ring. If K is a sufficiently large extension of Q
... finite group G, a prime number p and a finite extension K of the p-adic field Q p such that KH is split for all subgroups H of ...in K, by p the maximal ideal of O, by k the residue ...
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For example, suppose k to be a p-adic field and G to be the multiplicative group k×
... of G with the same central ...a commutative ring with unit, in proving that a module is projective if and only if it is a summand of a free ...not commutative and does not possess a ...
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The paper devoted to implementation of the public key algorithm based on directed algebraic graphs over finite commutative ring K and their symmetries
... map. Let us consider the case t j = kl, where k is a small number and the sequence α 1 , α 2 , ...period k and the transformation h = u −1 N α 1 N α 2 . . . N α k u is known for Bob in the ...
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The k Zero Divisor Hypergraph of a Commutative Ring
... a commutative ring has been studied by many authors, and the k-zero-divisor hypergraph of a commutative ring is a nice abstraction of this ...interesting. Let R be a ...
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COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
... A commutative ring R is a domain if and only if the product of any two nonzero elements of R is nonzero... the can- cellation law holds. We should prove that the product of any two nonze[r] ...
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Transparency of Polynomial Ring Over a Commutative Noetherian Ring
... Noetherian ring R is said to be transparent ring if there exist irreducible ideals I j , 1 ≤ j ≤ n such that ∩ n j=1 I j = 0 and each R/I j has a right artinian quotient ...ring. Let R be a ...
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Strict Units of Commutative Ring Spectra
... • K 0 M (2) ∼ = Z p is generated by γ = P b∈F × p χ (b,0) − χ (0,b) ...Atiyah’s definition of the transfer in K-theory, which in algebra corresponds to induction of ...
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Prime Graph of the Commutative Ring Z
... 2 Let us take the ring Z 125 , the non-zero elements which are adjacent in P G(Z 125 ) are {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, ...
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On the genus of the annhilator graph of a commutative ring
... that K 3,6 is a subgraph of AG(R). By Lemma 3, g(K 3,6 ) = 1 and hence one can fix an embedding of K 3,6 on the surface of ...of K 3,6 , say {S 1 , . . . , S 9 }. Let n i be the ...
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The compressed annihilator graph of a commutative ring
... 2017) Let R be a commutative ...reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of ...
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A Note on a graph associated to a commutative ring
... undirected. Let G = (V, E) be a graph. Let a, b ∈ V, a 6= ...in G between a and b if there exists such a path in G; otherwise we define d(a, b) = ...of G, denoted by ...
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A graph associated to spectrum of a commutative ring
... Proof. By using the first isomorphism theorem for rings, R/Ker(f ) ∼ = R 0 . The result now immediately follows from Theorem 2.1 (i). Remark 2.3. Let R 0 be an extension ring of R and q ∈ Spec(R). A prime ...
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Abstract. Let R be a commutative completely primary finite ring with the unique maximal ideal J such that J
... Abstract. Let R be a commutative completely primary finite ring with the unique maximal ideal J such that J 3 = (0) and J 2 6= ...p k , where 1 ≤ k ≤ 3, for some prime p and positive ...
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Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring
... Abstract. Let R, S be rings with unity, M a module over S, where S a commutative ring, and f : R → S a ring ...A ring representation of R on M via f is a ring homomorphism µ : R ...
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Edge Sum Index of a Graph in a Commutative Ring
... Theorem 1.3. Let Γ(Z n ) be a edge mapping f : E(Γ(Z n )) → Z + and edge sum function F. If v is the only vertex such that F v / ∈ Z + , then v is adjacent to a pendent vertex. Proof. Let deg v = m and v 1 ...
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Number theoretic properties of the commutative ring Zn
... Definition 2. Let 𝑎, 𝑏, and 𝑚 be any positive integers. If 𝑛 divides the difference 𝑎 − 𝑏, then we said that 𝑎 is congruent to 𝑏 modulo 𝑛 and we write 𝑎 ≡ 𝑏 (mod 𝑛) . In other words, it is equivalent to the ...
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A REPRESENTATION OF RING HOMOMORPHISMS ON UNITAL REGULAR COMMUTATIVE
... semisimple commutative Banach algebra and B a unital semisimple commutative Banach algebra, which need not be ...a ring homomorphism ρ : A → B and give a representation of ρ; hence a description of a ...
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Definition. Let V ⊆ A
... a k t k for some k ≥ 0 and for suitable coefficients a 0 , a 2 , a 3 , ...a k ∈ K. u and u − a 0 generate the same ring, so we may assume that a 0 = 0 and u = a 2 t 2 + a 3 t 3 + ...
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THE LINE GRAPH ASSOCIATED TO THE TOTAL GRAPH OF A COMMUTATIVE RING
... of G, then γ(H) ≤ ...graph G is planar if and only if it does not contain a subdivision of K 5 or K 3,3 .... Let us give a brief review of known results concerning the genus of complete ...