पारिवारिक व्यवसाय से आप क्या समझते ह? पारिवारिक व्यवसाय में संघर्ष के कारण बताइए।
पारिवारिक व्यवसाय से मेरा मतलब वह व्यवसाय है जो परिवार के सदस्यों द्वारा मिलकर संचालित किया जाता है। इसमें परिवार के सदस्य अपने संसाधन, समय और मेहनत को जोड़कर व्यवसाय को सफल बनाते हैं। यह व्यवसाय अक्सर पीढ़ी दर पीढ़ी चलता रहता है, जिसमें अनुभव और ज्ञान परिवार के सदस्यों में हस्तांतरित होता है। ______ _______ _______ _____ ________ ________ ______ _______ ____ ________ __________.
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(i) Give, with justificaiton, a nilpotent element in R.
(ii) Give, with justification, a zero divisor in 푅 which is not nilpotent.
(iii) What is the order of U(R)?
Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x³ +12) is a field.
Find the order of each of the elements in U(15). Is U(15) cyclic? Justify your answer.
Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.
Check whether R is a subring of
? Justify your answer
Define an abelian group. Give an example of a non-abelian group. (You don't need to prove that your example is a group. You have to only prove that it is non-abelian.)