Ring theory MOC

Centre of a rng

The centre Z(𝑅) of a rng 𝑅 is the subrng consisting of all elements of 𝑅 that commute with every other element, i.e. Z(𝑅) ={𝑎 ∈𝑅 ∣𝑎𝑥 =𝑥𝑎 ∀𝑥 ∈𝑅}. ring This is entirely analogous with the centre of a group.

Properties

• The centre is necessarily a commutative ring
• If 𝑅 is a ring then 1 ∈𝑍(𝑅)

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