# The Center of a Matrix Ring [duplicate]

Prove that the center of the ring $M_n(R)$ is the set of scalar matrices. I know what a center look like but i feel like i have not enough information to even solve this problem. Anyone that can help that would be great.

## marked as duplicate by Michael Albanese, rschwieb ring-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Feb 5 '15 at 20:31

• Double containment is the best way to do it. Let $Z$ be the center and $D$ be the set of scalar matrices. Certainly scalar matrices are contained in the center, as $A(\lambda I) = \lambda (A I) = \lambda A = (\lambda I)A$. The other direction is a little harder. Try direct computation of a non-scalar matrix. – walkar Feb 5 '15 at 20:31