A **group ring** is a mathematical structure that is used in abstract algebra, combining concepts from both group theory and ring theory. More specifically, if \( G \) is a group and \( R \) is a ring, the group ring \( R[G] \) is a new ring constructed from these two objects. ### Construction of the Group Ring 1.
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The group ring consists of finite formal sums with coefficients in a ring . Addition is coefficientwise and multiplication extends the group operation and multiplication in distributively. A group algebra is the case in which is a field.