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 Incoming links: Unit group of a direct product of rings

Past exam of the rings and modules course of the University of Oxford / 2023 / 1 / a / i / Solution 2026-10-25

Multiplication in a direct product of rings is coordinatewise. Thus

(r_{1}, \dots, r_{n})

has a two-sided inverse exactly when each

r_{j}

has a two-sided inverse in

R_{j}

, and then

(r_{1}, \dots, r_{n})^{- 1} = (r_{1}^{- 1}, \dots, r_{n}^{- 1}) .

Consequently its unit group is the direct product of groups

U (R_{1} \times \dots \times R_{n}) = U (R_{1}) \times \dots \times U (R_{n}) .

The identification preserves multiplication because both sides use coordinatewise multiplication. This is the unit group of a direct product of rings formula.

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