Decimal representation

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

Motivation

Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring R has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring.

Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories are equivalent in the sense of category theory.

This notion is of interest only when dealing with noncommutative rings, since it can be shown that two commutative rings are Morita equivalent if and only if they are isomorphic.

Definition

Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.

Examples

Any two isomorphic rings are Morita equivalent.

The ring of n-by-n matrices with elements in R, denoted M_n(R), is Morita-equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if M is a left R-module then Mⁿ is an M_n(R)-module where the module structure is given by matrix multiplication on the left of column vectors from M. This allows the definition of a functor from the category of left R-modules to the category of left M_n(R)-modules. The inverse functor is defined by realizing that for any M_n(R)-module there is a left R-module V and a positive integer n such that the M_n(R)-module is obtained from V as described above.

Criteria for equivalence

Equivalences can be characterized as follows: if F:R-Mod $\to$ S-Mod and G:S-Mod$\to$ R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that _SP and P_R are finitely generated projective generators and there are natural isomorphisms of the functors $\mathrm{F}(-)\cong P{\otimes }_R-$, and of the functors $\mathrm{G}(-)\cong \mathrm{Hom}{(}_{S}P,-).$ Finitely generated projective generators are also sometimes called progenerators for their module category.

For every right-exact functor F from the category of left-R modules to the category of left-S modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that the functor $\mathrm{F}(-)$ is naturally isomorphic to the functor $E{\otimes }_R-$. Since equivalences are by necessity exact and commute with direct sums, this implies that R and S are Morita equivalent if and only if there are bimodules _RM_S and _SN_R such that $M{\otimes }_{S}N\cong R$ as (R,R) bimodules and $N{\otimes }_{R}M\cong S$ as (S,S) bimodules. Moreover, N and M are related via an (S,R) bimodule isomorphism: $N\cong \mathrm{Hom}(M_S,S_S)$.

More concretely, two rings R and S are Morita equivalent if and only if $S\cong \mathrm{End}(P_R)$ for a progenerator module P_R, which is the case if and only if

$S\cong e{\mathbb{𝕄}}_n(R)e$

(isomorphism of rings) for some positive integer n and full idempotent e in the matrix ring M_n(R).

It is known that if R is Morita equivalent to S, then the ring C(R) is isomorphic to the ring C(S), where C(-) denotes the center of the ring, and furthermore R/J(R) is Morita equivalent to S/J(S), where J(-) denotes the Jacobson radical.

While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic. An easy example is that a division ring D is Morita equivalent to all of its matrix rings M_n(D), but cannot be isomorphic when n > 1. In the special case of commutative rings, Morita equivalent rings are actually isomorphic. This follows immediately from the comment above, for if R is Morita equivalent to S, $R=\mathrm{C}(R)\cong \mathrm{C}(S)=S$.

Properties preserved by equivalence

Many properties are preserved by the equivalence functor for the objects in the module category. Generally speaking, any property of modules defined purely in terms of modules and their homomorphisms (and not to their underlying elements or ring) is a categorical property which will be preserved by the equivalence functor. For example, if F(-) is the equivalence functor from R-Mod to S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple, semisimple, finitely generated, finitely presented, Artinian, and Noetherian. Examples of properties not necessarily preserved include being free, and being cyclic.

Many ring theoretic properties are stated in terms of their modules, and so these properties are preserved between Morita equivalent rings. Properties shared between equivalent rings are called Morita invariant properties. For example, a ring R is semisimple if and only if all of its modules are semisimple, and since semisimple modules are preserved under Morita equivalence, an equivalent ring S must also have all of its modules semisimple, and therefore be a semisimple ring itself.

Sometimes it is not immediately obvious why a property should be preserved. For example, using one standard definition of von Neumann regular ring (for all a in R, there exists x in R such that a = axa) it is not clear that an equivalent ring should also be von Neumann regular. However another formulation is: a ring is von Neumann regular if and only if all of its modules are flat. Since flatness is preserved across Morita equivalence, it is now clear that von Neumann regularity is Morita invariant.

The following properties are Morita invariant:

• simple, semisimple
• von Neumann regular
• right (or left) Noetherian, right (or left) Artinian
• right (or left) self-injective
• quasi-Frobenius
• prime, right (or left) primitive, semiprime, semiprimitive
• right (or left) (semi-)hereditary
• right (or left) nonsingular
• right (or left) coherent
• semiprimary, right (or left) perfect, semiperfect
• semilocal

Examples of properties which are not Morita invariant include commutative, local, reduced, domain, right (or left) Goldie, Frobenius, invariant basis number, and Dedekind finite.

There are at least two other tests for determining whether or not a ring property $\mathcal{𝒫}$ is Morita invariant. An element e in a ring R is a full idempotent when e² = e and ReR = R.

• $\mathcal{𝒫}$ is Morita invariant if and only if whenever a ring R satisfies $\mathcal{𝒫}$, then so does eRe for every full idempotent e and so does every matrix ring M_n(R) for every positive integer n;

or

• $\mathcal{𝒫}$ is Morita invariant if and only if: for any ring R and full idempotent e in R, R satisfies $\mathcal{𝒫}$ if and only if the ring eRe satisfies $\mathcal{𝒫}$.

Further directions

Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist for subcategories. In other words, because infinite dimensional modules are not generally reflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings. Perhaps not surprisingly, the criterion above has an analogue for dualities, where the natural isomorphism is given in terms of the hom functor rather than the tensor functor.

Morita Equivalence can also be defined in more structured situations, such as for symplectic groupoids and C*-algebras. In the case of C*-algebras, a stronger type equivalence, called strong Morita equivalence, is needed to obtain results useful in applications, because of the additional structure of C*-algebras.

Significance in K-theory

If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of the classifying space of the nerve of the category of projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Template:Cite web